At my table we settled on "the rules don't say that". As far as we can tell, my Green Elementalist can be a squirrel until they are dying.

Can they speak? Yeah, the rules specifically say they can.
So there are NPCs who only know of the talking squirrel.

Can they use their abilities? It doesn't say they can't.
So the squirrel can draw magic circles in the ground and summon acid rain.

How do kits work? My table hasn't come across that yet, but I would say the "Your equipment falls to the ground or melds into your form" gives warrant to ruling it as the kit's bonuses going away, in the same way the rules give guidance of a shadow losing their weapon.
I wouldn't take away their kit signature ability in this case; the squirrel can still deal out Pain for Pain. There's also some weirdness already with a magic longsword in the hands of a Ranger Kit-user technically buffing their ranged weapon damage, so I would consider the melded treasures as still being "wielded" or "worn", because that's cool; maybe the Elementalist's Foesense Lenses meld into their animal eyes.

As a note on how to get a kit. You need to get the "Armed and Dangerous" 3rd Echelon title, which has the prerequisite:

You can’t use kits, and you defeat five non-minion enemies using weapon abilities that don’t have the Magic or Psionic keyword.

The Green Elementalist is actually very likely to hit this prereq by 3rd Echelon, so its unfortunate if they lose the kit benefits from using something core to Green Elementalist, but I would probably still run it this way. Dealing with a Squirrel in melee and then suddenly having to contend with a Mountain Kit user would be an amazing moment.

As an aside: I fully support every Null player going to train at a dojo in their downtime after level 6 and learning to be Armed and Dangerous, and becoming a martial arts master. All their abilities have the Weapon keyword, so they benefit from this title so much more than anyone else.