5e D&D has some personality characteristic mechanics in it. They basically boil down to "gain inspiration".
Double down on it.
I'd also recommend using "gritty rests"; short rests are overnight in a secure place, and long rests take a week in a secure place. Sleep is still required.
To be generous, allow expending hit dice to regain a level of exhaustion (roll a 5+, including constitution modifier) during a short rest. You could also allow them to roll all of their expended HD from before the rest and recover everything that lands on a 6+ if the rest is particularly good.
Then wilderness challenge is going to be a game of attrition and attempts to secure rests. Inspiration - the ability to get rerolls - should be key.
Have some checks, where failure has consequences. Allow rerolls by either inspiration (where the reroll is with advantage) or by taking a level of exhaustion.
Use the D&DOne exhaustion - each level of exhaustion causes a -1 penalty on all d20 tests. Add in throw in -10% max HP per level and -10% speed for more fun. Death occurs at level 11.
Gage each overnight rest:
- No rest: you suffer 1d4 levels of exhaustion
- Poor rest: you suffer a level of exhaustion
- Minimal rest: you don't get a level of exhaustion
- Good rest: You gain the benefits of a short rest
- Excellent rest: You can expend a HD to regain a level of exhaustion on a 5+ (including con bonus), or expend a HD to regain that much ability damage.
- Amazing rest: Roll all expended HD from before the rest. Any even rolls 4 or higher are regained.
Keep track of supplies. Each day you ideally require 5 pounds of food and 5 pounds of water (I picked 5 as it is a round number and close enough) when reasonably active, moreso if you are more active. Hunted animals can be turned into food at a rate determined by a survival check.
Your ability to store food and water will be limited; so you'll constantly be trying to find food and water. When you are short, you'll be making constitution checks to avoid exhaustion.
All of this is providing the mechanical framework. The goal is to make the player really really want those rerolls that inspiration gives.
By making inspiration a reroll with advantage, and making the rolls be relatively rare but high-stakes, it means that the player roleplaying their character between rolls becomes important.
I'd allow the player to keep more than 1 inspiration, to avoid "why would I roleplay when I already have one", but if you already have inspiration you have to pass a check to get more; say, a flat d20 roll against (10+number of inspiration you have).
That will prevent building up an infinite stockpile, while also rewarding continued "self roleplay".
Instead of calling for a check on every action, just keep track of if the character is trained and if the player's plan is solid. Narrate successes. (ask for rolls more often on untrained stuff).
Only once in a while require a check after a complication happens. The goal is that any check you make has a significant failure consequence. Your goal is to tell a story, not roll d20s.
"You travel for 3 days and make it to the other side of the grasslands. Finding small game has been easy, but there hasn't been much water; your water skin is almost empty" for example. No check really needs to be made, except now the player needs to find water. The stakes are higher than a daily "did you find food" check.
Have enough rolls that the player doesn't get to keep an insane bank of inspiration. You can actually keep track of how high the player has stacked their inspiration and respond with a more complex problem to be dynamically evil.
I'd attempt to time box the travel time in real-world pacing. Decide up front how many sessions it will involve. Change your mind based on how fun it is working out. Have a combat encounter in your back pocket to "spring" on the character to eat up time if your "travel" mechanics aren't turning out to be fun, and revise next session.