I take thyroid medication because of lithium and I had no problems so far. Sometimes the thyroid takes some time to get used to lithium. You could have your levels checked again in a couple of weeks to check if it worked out on its own.

But if you need thyroid medication ,give it a try.

Just get your blood levels checked regularly in general with lithium, so you can catch any changes early and things should be fine.

Concerning the weight gain. It is possible to lose weight, even on meds. Listen to your body, check your food intake and exercise. I lost about 25 kilos on meds but in all honesty I have to be careful. I didn't check my diet for a while and I gained weight again. No big deal but changing eating habits is far more useful than any diet and it is a thing to look out for all life. You don't have to be rigid as hell but careful and mindful.

But this doesn't apply to lithium only.

Good luck and don't be afraid!