Oh my Rh what a weird turn & yeah I would be mad to at the turn of things. Hope the new vet can get your cat well. Seems to me he swallowed something - that makes more sense than anything else.
Here is my history on putting animals to sleep - just a FYI. 1st dog I put to sleep was Ben, he was probably about 13, Oct 31st. Took him late night to er & they said he had a heart attack & was in congestive heart failure. He screamed the entire drive. They recommended I put him down - I did - vet was very sympathetic & nice. People I know / knew said never listen to that advice - they could've cured him, you should've brought him home & etc. etc. So next super sick pet was Lizzy our cat - didn't put her down - she died naturally at home & IT WAS THE WORST! My friend warned me about it - said you don't want to see that! Poor thing suffered but worse than that she some how thought I was her enemy at the end & gosh NOT so - just freaked out I guess & death is hard to face - I guess.
Put my 18 yr old cat to sleep & it was a big relief, he was supposed to be on IVs prescribed by the vet but he hated them so - I let him have his time in the sun on the deck & then when he was too sick put him to sleep. He hung on for sometime.
Gotta say though every pet before they left me did something weird like you said your cat did - strange look - even Max before he became a pancake in the road - last seen - weird look - looked sad.
Anyways - SO sorry - hope it works out 4 u & you're family.
all this; stay safe op and do the right thing. do your due diligence but when its time to go for the pet its time to go, don't let it suffer.