The policy from BA

"A sports racket for games such as tennis, squash or badminton is permitted as hand baggage if carried in a slim protective case, as one of your two pieces of hand baggage allowed in the cabin, providing:

• The case contains a maximum of two rackets and is under 80 cm long, 45 cm wide and 10 cm deep.
• No additional items, such as balls, shuttlecocks or clothing, are packed in the racket case.
• The racket case is stowed on top of other items in the overhead lockers to prevent damage and maximise space.

If you carry three or more rackets, these must be packed in a protective bag and checked into the hold as part of your checked baggage allowance."

The main issue I see with a string bag is that it wouldn't provide much protection for the rackets.

One thing to also consider is that the pressurisation in the hold is exactly the same as the passenger cabin. It is likely to be a little cooler though - typically around 5-10C on longer flights so it is likely the temperature not the pressure that you notice with the strings.

I would also suggest if travelling to / from tournaments to just get the rackets restrung on arrival - at least that way they'd be exactly how you'd like them.