My cat has hyperthyroidism. What are signs his thyroid is controlled?

I am sorry to hear that you and Ricochet are having a hard time. Unfortunately, the only way to see if the dose needs to be increased is to do bloodwork to evaluate his thyroid level. It is important to remember that there are other possible causes for him to be yowling at night, besides hyperthyroidism, including behavioral causes, cognitive dysfunction and potentially other metabolic causes and that increasing his methimazole too much will be harmful to Ricochet. I would recommend continuing the dose as prescribed by your vet and talking to them about what can be done to get Ricochet in for a blood test. Some clinics will accept payment plans or work with you to make sure that Ricochet gets the care he needs without putting you guys out of house and home. Again, so sorry you and Ricochet are going through this - you clearly care a lot about him and I hope he feels better soon!
Also, to answer your question, the signs you may notice will be resolution of his signs of hyperthyroidism (typically, hyperthyroid cats exhibit weight loss despite voracious appetite, increased thirst, increased urination, among other possible signs). Hyperthyroidism can also affect other organs in the body like the kidneys, heart, brain and eyes, so sometimes we can see secondary disease in these organs with hyperthyroidism.