Karl Friedrich wouldn't need to make the same agreements as Ulrika did. She had to make those agreements in exchange for the support that those nobles would give her against Karl Friedrich. However, Karl Friedrich, as the senior legitimate heir, wouldn't need to make the exact same agreements. Especially if Carl XII gets it enshrined in law that his Holstein nephew is to be his successor. If not, there's still enough people who might prefer Karl over Ulrika Eleonora (who has her sex (even though Sweden has had a queen-regnant before) and her childlessness working against her). Another reasoning, later applied, was that OTL Pyotr III (Karl Friedrich Ulrich) couldn't inherit the Swedish throne after Friedrich of Hesse because of his ties to Russia. Sweden used the uncertainty of the Russian succession as a reason to block Pyotr's candidacy (Yelizaveta Petrovna didn't really help matters, since by the time the Swedish delegates came to offer the crown to Pyotr, she'd already named him as heir to Russia (even though she knew he would be the first logical choice).
As to Karl Friedrich, if he succeeds as TTL Carl XIII, the chances of him marrying a Russian grand duchess is slim to none. Anna and Yelizaveta Petrovna weren't considered highly abroad for the ambiguous status of their birth. As duke of Holstein marriage to Anna is fine, as king of Sweden it's unthinkable. Which leaves him needing a wife (one of Friedrich the Great's sisters perhaps (there had been the agreement that Friedrich Wilhelm I would marry Ulrika Eleonora, but when that fell through, it was transferred to her son and his daughter)?) or somewhere else?