Question: I'm going to be recording some piano music in the near future, and I'm interested in the (possibly crazy) idea of tuning the piano to a non-equal-tempered system for the recording.

I am recording songs in F (x3), dm, fm, D (x2), and em. Would this require retuning the piano to center the tuning system around the key of each song? 

In the case of just intonation, I'd assume so. But is well-temperament key-agnostic, or would I tune the piano to "F well-temperament" and then "D well-temperament"?

(Bonus: Any idea how I'd find a tuner who could do this?)

Answer: 
The point of unequal well-tempered tunings is that the keys don't sound the same. Temperaments like Werckmeister III or Vallotti or 18th century French *ordinaire* are meant to be usable in any key while letting each have its own colour. (Vallotti is quite commonly used on fortepianos.)

For most of what you're doing, you could probably get away with a mean tone tuning (which isn't well-tempered, but does a reasonably good job with the keys around C Major - it was the original compromise), but F minor would likely be problematic for the standard 1/4-comma temperament. A 1/5-comma or 1/6-comma tuning *might* do the job.

All of these temperaments, well-tempered and mean tone, give the brightest character to C Major, with an increasing dark flavour as you move away by fifths.

To find a tuner, you'd best make use of any contacts you have in the Early Music field, and see if they know of piano tuners who deal with this. (Yeah, there are apps for tuning, but you'll probably want someone who can do it by ear, because on a modern piano, you'll also need the tuning *stretched* to deal with the inharmonicity of modern piano strings.)
*Equal* temperament is key-agnostic. *Well-tempered* tuning isn't. The point of well-tempered tuning is that all keys are tolerable, but some are still better than others and each has its own character.

This own character was pretty much the whole point of Bach's "Wohltemperirtes Clavier" (it is usually assumed that some Werckmeister tuning reflects the expected tuning best). Since such a tuning is supposed to lend different keys different characters, it would be quite pointless to retune to a different center of tonality for different pieces. Well-tempered tunings are a flavored *compromise*. If you can move a tuning around, there is no point in compromising. Just use just tuning then. If you can't move anything around, equal temperament is a flavorless compromise not distinguishing between keys.