Devils Advocate. If BA only reimbursed the actual expenses of living oversees then crew would only be "living" on their basic. For new joiners this is circa 10,000 pa dependant on base.
I am afraid that this is not going to attract many new employees to the role. Certainly not good ones. Moreover it would force out a lot of other people.
Basically the money saved would no doubt be lost in the demand for an increase in basic pay which, in this hypothetical case, the Cabin Crew would be more than justified in asking for.
The Allowance system is deeply complicated and probably needs a reform but the money it provides to Cabin Crew isn't really going to be saved as it forms part of a reasonable salary for many. (Not all Cabin Crew are raking it in).
Any savings would probably come from the simplification of the payment process.