I can't lie, Garp is receiving his flowers but when a named chapter introduction gets dropped you're not losing lol.
I think Garp will get wrapped up/subdued somehow, Aokiji basically said Marines won't move an inch for Koby - but they might for Garp.
Sounds ridiculous even typing the possibility of Garp taking some sort of L but it certainly wouldn't be his first. Guy is a has-been. His grandson is dead and all he did was say "hold me back" to Sengoku.
Garp taking L here makes sense.
As you said, he isn't Sword member so capturing him would fulfill the goal of BBP.
Second, it will fuel Luffy vs BB and could be a converging point
Luffy would mess with WG at egghead
BBP would mess with garp and sword
Luffy vs BB would become more personsal
So, a converging point for BB, marines and luffy