Getting a Jagellon on the Muscovite throne might be possible during the Time of Troubles,
There would be a tiny and quite insignificant problem: the last Jagellonian King of PLC, Sigismund II Augustus
, died in 1572, 26 years before the ToT started.
You are talking about Wasa dynasty and, yes, it was possible: Prince Wladislaw was officially acknowledged by what was passing at that time as Tsardom's government as a Tsar to Be but neither his father nor himself had been too keen on the precondition, conversion into Orthodoxy, and an idea of having a Catholic Tsar was absolutely unacceptable to the Russian side. Later Wladislaw's attempt to get throne by force (Wladislaw's Moscow Campaign) was doomed if not necessarily militarily (he got all the way to Moscow) then politically (Russia already elected its Orthodox Tsar).
but getting Muscovy into a lasting union or making the Muscovite Jagellons actually last would be even harder.
Religious issue would not go away on both sides.
In the case of the "Russian-centric" union (at some point, prior to the creation of the PLC, Ivan IV got some support of his candidacy among the Lithuanian nobility) there would be an additional problem: difference between absolutism and constitutional monarchy with a weak royal power. How to reconcile it?