Guys, wanna make it even more interesting? Let's say Ivan the Terrible's daughter from first marriage (either Maria (b.1552) or Eudoxia (b.1556) survives past childhood). Eudoxia is more preferable.
The Livonian war goes as OTL till roughly 1570, when there is no Interregnum and all this stuff.
The push of early 1570ies (including sack of Moscow by Crimean tatars in 1571) is strengthened compared to OTL. Depending on which kind of person our hypothetical Sigismund III is, 17-years old King may inherit very preferable situation, and enforce the peace treaty on Russians complete with dynastic marriage between himself and Eudoxia.
Flash forward 1590ies-1600ies - if Rurikids go extinct as OTL, the situation gets REALLY interesting, as children from Eudoxia are very important claimants to the throne.
Bonus points if the King does not alienate its Orthodox subjects thus removing propaganda card from Russian hands.