Vacuum, mist on RELEASIT EncapClean DS, hand wipe with micro fiber towel,force dry. It's not the felt you need to worry about, it's what's under the felt...
It's just felt adhered with contact cement, so any wet cleaning less than complete flooding will work fine. The only concern would be the very cheapest tables that use fiberboard instead of slate as an underlayment.
I'm not a super experienced cleaner or anything but my table has a 1" stone type slate under the felt. We've spilled everything on it and I don't think you can hurt the slate. I would be worried about causing the glue that holds the felt down to let go. The guy who assembled my table used some type of 3M glue in a aerosol can.
You're right Rick, a strong dry solvent will damage the adhesive. Wet cleaning won't. We use the same adhesive (3M Super 77) to attach backing fabric to custom fabricated area rugs and have no trouble with it when they come in for cleaning.
We did one that the cat was using for a litter box, Deodorized, let it sit cleaned with a Low Ph uph prespray and extract not a big deal. It's easy to be brave when they say fix it or it goes in the trash!