Some Beefing Up is Totally Reasonable
First off, the hand-crossbow, with its short range, loading property, and inferior damage to other ranged weapons is generally a sub-optimal weapon in most situations. It is effective in concert with a large supply of light throwing weapons in the other hand (load and fire crossbow, draw dagger, throw dagger: pretty decent for a rogue), and a few other situational times, especially if you allow preloading for true one-handed fire first round. But, generally it is suboptimal, and thus some beefing up can be more an instance of putting it on par with other weapons rather than outclassing them.
Taking that into account, let's consider the closest game analogues and how they will vary based on class.
The Closest Analogue For Use with Shields is Melee Weapons
Fundamentally when a warrior with martial weapons chooses a one handed melee set up they are making the decision for a less powerful primary weapon in exchange for a shield. Since for pure ranged weapons there is no existing shield option, and since for throwing weapons there is no two-handed option, pure melee seems to be our closest analogue. What the relative advantage is will vary based on what weapons are available to that class.
A class with martial weapons is generally making the decision of a two-handed weapon which will give them an average dice damage (not addressing crits) of 6.5 or 7 for a long word, or 5.5 for a polearm with reach, vs. having a shield or second weapon and having their primary damage do and average of 4.5. So a loss of 2-2.5 damage in exchange for the AC boost, or 1 damage and reach. Were they to use hand crossbow and shield in lieu of heavy crossbow it is a trade-off of 5.5 and vastly more range vs 3.5 with an AC boost, which is comparable and seems like a substantial and probably balanced sacrifice on this front.
While the advantages of more range are many, the particular shortness of the hand crossbow has an additional wrinkle. The 30 ft. non-disadvantaged range will mean that many enemies can get to you in one turn much of the time, and then you're a ranged fighter in melee, which for crossbow non-experts sucks.
The hand-crossbow would be comparable to use of throwing weapons with shield. This would then be less powerful by comparison for throwing, since you have to have a lot of extra weapons, but more powerful in terms of having a melee option, so again probably balanced.
Different Sort of Boost for Bards and Rogues
A light weapon one-handed weapon has a very different significance for shield-less classes that get hand crossbows, namely Bards and Rogues.
They don't have shields or two handed weapons beyond going two-handed with a long sword. In terms of the latter, the melee sacrifice they are making is generally between doing 2 more damage with a double handed longsword (and having to invest in strength to do it and not getting the rogue sneak attack damage) vs. something they can dual-wield and/or use with finesse. More often we have the trade between rapier, for a free hand and average increased main attack damage of 1, vs. short swords for dual wielding or daggers for dual wielding with a throwing option.
Beyond the earliest levels the dual wield options make less sense for bards as they increasingly have use for an instrument focus in the off hand and for the bonus action they would use the second weapon. For the rogue however, the choice of dual daggers or shorswords is much more powerful, as they get a second bite at the sneak attack apple.
What this really all boils down to, is that a rogue (and to a lesser extent the Bard) will benefit a great deal from the exchange between an average 4.5 and a lot more range with a light crossbow vs. 3.5 on a hand-crossbow, less range, and an offhand attack or otherwise free hand. Nevertheless the range difference is quite substantial. The hand crossbow requires being within movement range of most creatures to shoot without disadvantage and these are both light armor classes (depending on the Bard: Valor and Sword bards have different dynamics).
On the whole it still seems reasonably balanced as a choice for a bard: between having to stay fairly close to the threats with double hand-crossbows or with a hand crossbow and spell focus in hand, or getting to stand way back and having slightly stronger attacks, especially if your table is not sticklers about juggling that spell focus such that you can use it easily with a bow. The Rogue however, is really going to benefit from that extra attack. Or if the enemies rush them while ranged, they can dash or disengage with a bonus action so the danger from being a weapon wielder with a range of 30 feet from the enemy is somewhat less of a problem now.
The Elephant in the Room: Crossbow Experts in Melee
Where this all starts to, perhaps, unravel balance-wise is when you have people (either out of a particular aesthetic or because they are power-gamers) decide it is time to go handcrossbow and shield instead of melee weapon and shield in melee. Versus a one-handed weapon they would otherwise use it is the loss of one average point of damage for adding range to what they would otherwise do. That is probably objectively superior for anyone using a shield (or for our bards and rogues) to using an actual melee weapon. And then they could take the Sharpshooter feat and do the -5 to hit for +10 damage trade—that many people consider overpowered with a shield—which is otherwise impossible.
On this last point, however, I would say that they have to take a feat for it, namely Crossbow Expert. There are lots of other ways they can make their character awesome with that ASI, so this is a privilege they are paying for. Same goes again for upping the ante with an additional feat cost: Sharpshooter.
Sharpshooter
- Attacking at long range doesn't impose disadvantage on your ranged weapon attack rolls.
- Your ranged weapons ignore half cover and three-quarters cover.
- Before you make a ranged attack with a ranged weapon with which you are proficient, you can choose to take a -5 penalty to the attack roll. If you do so and the attack hits, it deals +10 damage.
Also, if we accept that the hand crossbow option is not all that much better amongst the ranged weapon options, then the same two handed vs. one handed melee math applies in this Crossbow Expert build vs. a normal two-handed Crossbow Expert build. If you don't believe that the feat as is has broken the game then this really isn't going to break the game for you.
Also the simple decision to hand out a magic shortsword or dagger (or anything else) of awesomeness rather than magic hand-crossbows can change the calculus of what is optimal at the DM's discretion.
Conclusion - Probably Balanced in Most Situations, Maybe Tweak
Basically this will make the hand crossbow a really strong choice for the rogue even without Crossbow Expert: one of the strongest, but probably not so strong as to be an issue.
It will make it a much stronger choice for everyone else than it already is (particularly bards spending on how stringent you are about holding magical foci), but it is weak as is so that won't be a problem, and it will have a comparable relationship to other ranged weapons for them as one handed melee has to two-handed melee.
But once you throw the Crossbow Expert ranged-weapon-in-melee-feature into the mix, it becomes much more problematic as suddenly it starts beating out melee weapons while still having range. But once again this combo really doesn't unbalance things any further than Crossbow Expert already does.
If it bothers you, however, one solution is simply to say that this feat's feature does not apply to these hand-crossbows. Maybe they have weird magic or design or maybe holding them steady with one hand while an enemy bears down on you is just harder than doing so with a two-handed ranged weapon.
Implementation
Any of the options you give would probably work fine, it really just matters whether you want it to be an anyone can do it thing, a you have to take a feat thing, or a find a specific item thing.
I would recommend number three where they are special items. It creates, I would say, the fewest balance issues (and especially vs. the panoply of other special items in the game). I've argued that it's really only in dual wielding by shieldless classes that the balance becomes questionable, and this would allow a DM to simply not let that rogue get ahold of two of them. It also generally provides the most flexibility to adjust a game element you seem on the fence about as it plays out.
The second suggestion is the least optimal for game balance, as it both increases the power of an already very powerful feat and creates the potential situation of having to retroactively adjust something that a player invested in (taking the feat). I would add that this option gives increased power to people exclusively. This is especially the case when they are using the shooting in melee feature, the lack of which being part of what would keep hand-crossbows balanced—even while having to stay in fairly close range.
