I don't eat or cook tilapia but I don't see why you can't cook each item sequentially at the prescribed temperatures.

Cook the vegetable dish first at 350 and and take it out of the oven when its done and cover it if it's not covered otherwise. Turn up the oven to 400 (maybe taking 5 minutes for the temp to rise) and then cook the fish. If the vegetable dish is covered, it will be plenty hot 15 minutes after coming out of the oven, when the tilapia is done.

Instead of baking the fish, you could cook it in a frying pan instead. When the tomato pudding is nearly done, sear your fish quickly on each side in a hot skillet. If you are concerned about the use of oil for frying, simply use a cooking spray instead. This is, of course, assuming your fish recipe can be adapted from being baked in the oven to prepared in a skillet.

Isn't cooking spray oil?

Yes, you're fine...just bake it longer...give it 15-20 and check for doneness...use an instant read thermometer to 145 or use a fork to make sure it's flaky and not transparent in the middle.

Baking the fish at 350° will dry it out. Increasing the temp to 375° is better but maybe only bake it 15 minutes. Keep a tight eye on in, these thin filets dry out easily.

You mentioned recipe for the fish so it may not be able to be fried. Elmer's suggestion is a good one as it allows the flavors to meld unless it is something that has contrasting flavors. An alternative is to place the dish in the oven when 20 minutes is remaining. If the fish flakes the way it should it is done. If not remove the other dish and cook it a little longer.

It also depends on the fish recipe. If it's supposed be crispy or something like that, then either do the fish separate or probably can be done on the stove as has already been suggested. If it's a recipe that can be slowed down, like a more moist/smothered or covered treatment, then putting it in at the same temp as the pudding is OK, but just add a few minutes to the fish cooking time as has been suggested.

Don't try doing both at a compromise of 375 if you haven't made the tomato pudding before and also aren't sure of adjusting time/temp on the fish.

I always bake fish in the toaster oven. Tilapia is good when cooked in a skillet. We fry it in butter with Cajun seasoning.

If you want to bake it at a lower temperature and longer, make sure the pan is covered. I use aluminum foil if the pan does not have its own lid.

It depends on the recipe you are making.

Isn't cooking spray oil?

Yes, I suppose it is.

Isn't cooking spray oil?

Yes, and often more to make the oil easier to spray. Like anti-foaming agents. Sometimes preservatives, sometimes stuff like flour or starches for baking. Check the can and see if it is just oil or not.

There are also pump-spray bottles of oil, or one can also just lightly oil their pan with a paper towel too.