OP’s wording suggests mesotrione is already in hand. Do you agree, mishmosh?

So my question is - which fertilizer do I use and after how much time after applying the Scott’s Starter Fertilizer?

Do you have Tenacity separately? And I don't recall from your previous thread(s), what is your lawn situation...new seeding? overseed? spring or fall seed?

If you already have Tenacity, you could use any fertilizer.

The initial application was spring seeding using Scott’s Starter to fill in some bare spots but going forward I will use Tenacity separately, so I wanted to make sure not to get fertilizer that also contains weed killer, is that the correct way to go?

I would avoid weed-n-feed products until your grass is mature (after several mowings). Regular fertilizer is fine. Another application of starter with mesotrione also is fine.