Removing your FF number from your BP isn't guaranteed to work.

I once had QF (who handle FJ check-in at BNE) remove my QF number and try to replace it with my AA number for a BNE-NAN-LAX flight last year. They were unable to do so, despite calling for a supervisor, since they couldn't apparently access the right part of FJ's computer system. So they pulled the QF number and printed an FF number-less BP for me. To no avail - the flights credited promptly to QF two days later and no amount of complaining would fix it.

I think even if they had got the AA number to print on the BP, it still wouldn't have credited correctly, since with three airlines to deal with, apparently there are multiple places for the FF numbers to show up and even when you think you've got them all fixed, another problem can pop up...

The simplest solution is never to switch FF numbers on a single booking. Provide one at booking time, and don't change it. Otherwise you're asking for trouble.