When regular F-22s were already used to shoot down Chinese spy balloons they employed AIM-9 missiles, which are cheaper than AIM-120. But there's no need to get particularly close: in fact for safety they want to remain well separated from the target and never get on a collision course.
But in general all of your comments here have been deeply uninformed about how this stuff actually works.
This is a low stakes brainstorming conversation, I presented my ideas as hypotheticals and shared my sources.
Honestly, I was hoping someone more knowledgeable would share any better insight on what the recently work on the SR-71 frame would be used for in practice. That hasn't happened so far, but that's fine.
"Your guess is bad because using the SR-71 for _____ in missions such as ______ would make more sense" would be stellar brainstorming about this beloved fan favorite.
But in general all of your comments here have been deeply uninformed about how this stuff actually works.