Here’s a video clip including both edits.

I got around the Greef situation by just cutting more. It’s not implicit that Greef is meeting Mando a long journey away from the city, so we don’t need travel scenes. Greef begins with his bodyguards, which is sensible even with this interpretation, since they did leave each other on bad terms. That’s the purpose they serve here, then we don’t need to see them again. Then I flipped the scenes so that Greef asks to see Grogu first - and the tension dissipating there as we see that he also genuinely cares for Grogu is what sells us on the fact that they’re now not going to get into conflict. Greef just seems legitimate. Then they talk about the situation a little, but Mando just says “I have a plan.” (I know you like it when we just get straight into the plan, rather than explaining it first!) We then have a quick montage, then just cut straight to Greef escorting Mando to the Imperials - if we notice Greef’s injury, we can assume he faked it to make it look like his capture of Mando was legit. A smooth musical transition helps sell this all too.

For the second scene, “You just got your bell rung, you’ll be alright” is a legitimate statement, rather than Cara being actually worried about his mortality. The flamer then immediately attacks, so we don’t waste any time with them forcing an emotional scene. Grogu works his magic, and IG opens the grate then picks up Grogu. Again the music then carries us to the scene of Mando walking - we don’t need a wipe. Mando updates them on the situation as he’s struggling, then he stops and tells them he can stand. He’s now OK, and has recovered from essentially just a temporarily disorienting knock to the head.

I don’t massively feel like that’s the turning point for him not hating droids - I always felt like that was more of an odd prejudice (plus IG is an assassin droid) that he just gradually gets over.