Ivan IV's candidacy was mainly just used as a political tool by the Lithuanian nobility to frighten the Polish side during the 1573 and 1576 elections, something like "if you try to elect a Habsburg for serious, then we're leaving the Union under Iwan". I'm not even sure if it even worked since, as you might imagine, that's a really ridiculous ultimatum (though Anti-Habsburgs won both elections in the end so 🤷♂️).
It's doubtful Ivan IV ever seriously considered standing for election, he did not send any envoys to the elections so he never actually stood as a candidate. His territorial demands were ridiculous and there wasn't a chance that he could have gotten a majority. This is still a good century before Russia can seriously influence the royal election - and as it was at war with the Republic in the 1570s his options were limited even more.
And, of course, only Catholics can get elected King, and if Ivan's candidacy developed beyond Lithuanian musings then this question would have been brought up instantly. He could offer his son Fyodor of course and get him to convert, not that it overcomes all the other issues in getting a Russian king on the throne in 1573 or 1576 but still.
Only way I can see it happening is if Russia achieves such a total and unbelievable victory in the Livonian War that there are Russian troops standing outside of the election field and Iwan decides that he wants the throne of Poland for himself. At that point, though, the scenario is changed massively.