Damage during ground ops normally means something has crashed into the aircraft. If this were an external contractor, then they "could" be off the hook (though I'd argue against that). If it were damage by a company not involved with BA (e.g. an AA truck going to an AA aircraft hitting BA) then fair enough. If it is BA themselves then they are IMO liable.
In this case it seems like a simple technical issue rather than anything crashing into anything, in which case BA is liable.