If it's a small bird bedded down quietly under the jumpseat, I don't think I'd worry about it too much. Presuming that you found out about it while up at altitude, the question is, are you safer cruising for XX minutes to get to your destination, descend, and land, or are you safer cruising for a shorter time while you return to your departure airport, descend, and land? Either way, the bird is in the cockpit for a descent and landing somewhere, and if it seems to be quiet & unobtrusive, I don't see significant risk in the extra time at cruise.
On the other hand, if the bird is wildly flapping around & being an annoyance & a distraction, then minimizing the time that you're putting up with that would absolutely be worthwhile. Even small birds have enough beak & claws to scratch skin, and I wouldn't trust a scratch from any bird not to become a nasty infection.
I think something like this falls entirely within the realm of captain's best judgement -- I've never seen any written guidance at anything approaching the level of "consideration for a bird in the cockpit (discovered after takeoff)".
I'm also drawing a blank for anything similar in recent history, although I suspect that somebody somewhere has probably seen something pretty close. When the major U.S. airlines log multiple millions of flights each year, even the really rare cases tend to come up now & then.
As for "just kill the bird," I suspect that the hard part of that plan would be in the execution of it (the plan, and the bird too, I guess). If it's a fly buzzing around the cockpit, a good smack with the paper copy of the dispatch release will do the job, but for even a small bird, it will take more than that -- which tends to have its share of risks when wielding small improvised weapons in a confined space full of glass screens & instruments, circuit breaker panels, and another pilot whom you'd really like to avoid injuring!
It would be tempting to try to shoo the bird out of the cockpit -- either into the passenger cabin (which has its own concerns -- how long do you leave the door open waiting for the bird to fly out), or opening a cockpit window (at low altitude -- you can open them with sufficiently low pressure differential) & getting it out that way. Of course, in an MD-90, you might have just given yourself a bird strike in an engine that way, so maybe just landing to deplane the fowl passenger would be the better option!