How are you giving your cat the med? Is he throwing it back up after he
swallows it, or is he coughing it up before it gets down his throat?
Cats are averse to swallowing pills, and normally spit them up without
the pills getting into their stomach.

If you don't have a pill gun, the only way I know how to pill a cat is:

1. Lay the cat on his back upon your lap.

2. Hold his mouth open. Tilt his head back so that he can't completely
close his mouth.

3. Drop the pill into his mouth so that it goes into his throat.

4. Stroke his throat with your finger. This engages the swallow reflex
so that he has to swallow the pill.

If you've done all this already and he still won't take the pill, don't
give up on treatment yet. The SPCA and certain shelters and humane
societies provide treatment for animals at charitable rates.

Before you put him down, stop with the medication, and see how well he does
without it. IOW, give the cat a chance.....

if the cat needs treatment, and you can not provide it for him, then the
humane thing to do would be to try to find a new home for him.

The vet wouldn't give you medication that was "too high" for him.  If you
told him you couldn't afford testing, then chances are he gave you a small
dose to try to help out with out doing damage of an overdose.  I am assuming
it is tapazole or methimazole (same drug different name) in a 5mg dose?
once or twice a day?  My cat was started out on 2.5 mg once per day, but a
different vet went straight to 5mg twice a day.  she's now on 7.5mg twice a
day.

After pilling, you should immediately offer food or water to help the pill
go down the throat.  Just as you take pills with water, so should pets.  If
not, the pill gets stuck in the throat and causes problems - and yes,
vomiting is one of them.

Since he is still losing weight, the thyroid is still out of control.  He is
not getting enough nutrients out of the food.  I am sure he is very
uncomfortable, so the hiding is normal.

Please please please reconsider the testing, and treatment.  If your
finances truly can't afford it, and you can't find help, your poor kitty
will starve to death despite having food all around.  It is a hideous and
painful way to die.  He does not need to be put down, at 13 he still could
have a number of good years.