Assuming you are doing a routine tag fix that doesn't require discussion...
How many posts should someone re-tag en masse (which bumps messages in the
active tab) before asking a moderator to use their non-bumping, tag-renaming super powers?
I'd say at whatever point posting the request here on meta becomes less work than manually retagging the posts. Somewhere around 10 but less than 20 is my rough guesstimate. Any more than that would definitely draw complaints that the tag was being spammed (which I know from past experience).
I don't think you should worry about question bumping, it would probably do them good to see the light of day one last time.
For myself, I wouldn't think twice about 10, but would probably balk at 50.