A reader in the comment section in one of the recent posts below included a link to some good information about hyperthyroid condition in cats, and yes, it does mention the vomiting that can be a side effect of the medication.
I spoke to the vet yesterday, and she said Hamilton's thyroid tests had come back in the high normal range this time, showing that the medication was likely working. Now the trick is to be able to keep him on it without him throwing up. He has been keeping his food in fine for the past few days, so we've started him back on half doses of meds twice a day, and we'll see what happens.
The vet said that if we can keep him on the meds for three to four weeks, they can then do some tests to determine if his kidneys will be strong enough to handle the iodine irradiation treatment. Our ultimate goal is to get him that treatment if it won't endanger him. Poor guy! He's snoozing on the floor right now. His coat and weight do seem improved, though, don't they?