Why doesn't Latin caseus from *kwh₂et- have "w"?
An intermediate *quaseus is indeed hypothesized. Fay calls to compare with Plautus' cassat=quassat (Lewis and Short: quasso (old form casso, Plaut. Bacch. 2, 3, 71 Ritschl)).
He says (Fay and Ed, Greek and Latin Word Studies, The Classical Quarterly, Vol. 1, No. 1, 1907):
The equation of caseus 'cheese' with O.B. kvasû 'fermentum' may be correct, but no valid inference, I think, can be drawn from it for the treatment of kw-. It does not demand discussion at this time when I am seeking to establish two negative conclusions, (I) that there is no good evidence to prove that *kw- yields Greek k, Lat. v-; (2) and no good evidence for k̂ w- yielding Lat. v-; but I suspect that Plautine cassat=quassat gives us a right to regard caseus *quaseus. I have small doubt that cāsseus (older form cāseum, neuter) is derived from from *lac quassum, with suffix like the suffix of cereus and farreus. ...
It's not at all certain that Lat. cāseus "cheese" derives from some such PIE root as *kwath₂- "boil" (this is the form of the root given in LIV, though both it and your *kwh₂et- look rather odd), precisely because of the phonetic difficulty you mention. De Vaan in his etymological dictionary does not accept the connection, and concludes that the etymology of cāseus is unknown.