Did the other five actually want to travel without Mum?
Offloading bags belonging to FTJs is a security precaution, to try and prevent someone putting a bomb in a bag and then staying in the airport lounge while it gets airborne and goes bang, right?
How does this precaution now figure in the days of suicide bombers etc.? It is still relevant in that we don't need to make it easy for terrorists, but will never be able to make our security airtight.
If family are determined to bring on a bomb and 'mother' is willing to sacrifice them to achieve that, then the fact that one bag 'belongs' to her probably makes no difference to the odds of that one being the one with the bomb in or not - why not ask if there is a particular bag she wants offloaded for her convenience, rather than security reasons?
If we can take rush bags without an accompanying pax, then if normal pax bags have been screened to the same standard (i.e. not in France, only 50% of bags x-rayed to my knowledge) then why not take them with their accompanying 'family'?
If they are intent on suicide bombing the flight and have got their IED past security then offloading one bag ain't gonna stop 'em anyway!
A quick phone call to the airline security manager should cover your arse in the meantime.