Dunno what glue is good, but you might want to check out "archival" tapes and adhesives from framing shops. I'm no expert, but I'd think that the outgassing concerns would be mostly the same. Personally, I'd use the same Super77 spray adhesive that Ken recommended. If you're worried, you could let it dry for a few days outside of the box — I'd think any outgassing would be worst during the first stages of curing (but again: not an expert).
With respect to the fun of getting it in place I can recommend the "half and half" technique:
Cut the felt to size and lay it smooth and properly aligned on the bottom before putting any glue down. Then keeping, say, the left half down and aligned on the bottom, fold up the right half and lay it over the left. Spray the glue down on the right, then carefully unfold/roll-out the part you folded up. The unglued left half will ensure you still have the edges aligned. After the right side is firmly attached, fold up the left half and repeat the process.
Any other technique will have you pulling your hair out unless you use extremely slow setting glue.
It's still worth going through a dry run or two to figure out how to roll each half onto the adhesive while avoiding stretching the material and keeping it aligned to the edges.
Regards,
--
Rex