@goldenmammed
Okay so!! Here’s what I came up with!
Option 1) Perhaps Bezi is among the Starks + their bannermen during Jaime's captivity by the Starks? Maybe he replaces Brienne in the whole escorting Jaime to... (It's been a couple years since I watched the show forgive me for inaccuracies) Tyrion in exchange for the girls?
Knowing of [Jaime], Bezi would have an interest in Jaime and, being [formerly?] loyal to the Starks, he probably would have been been around Jaime a good bit; mostly making sure neither Jaime nor anyone else did anything funny. Bezi wouldn't be happy about the Bran situation but he has this. Infatuation with Jaime.
If he's put in Brienne's position he would have her sense of loyalty to Catelyn and be just as. Insistent about the goal and getting Jaime where they need to go. He'd probably be a bit light-hearted about the matter, tho, at least at first.
Option 2) In your verse where Jaime becomes king, perhaps Bezi somehow ends up close to him? Bezi's quite a skilled swordsman and could have lots of opportunities to bump into a king.
He would probably have a bit of admiration for Jaime, at the very least. Perhaps he somehow helps Jaime out? Perhaps aiding against an assassination attempt or something?
Option 3) At some point in Jaime's adult life but before Ned is killed, maybe Bezi and Jaime befriend each other? Perhaps at a tourney or on some other occasion where their skills as fighters are pitted against each other. Maybe they're so evenly matched that it's laughable?
Option 4) I'm bad at plots sorry