there are rooms available at the Premier Inn T4 and T5 on their website for tonight so not sure why you think it’s not available. Do you mean there are
no family rooms available?

Sofitel is £306, I suspect BA likely to only pay up if you can demonstrate no other option, and not based on confirmed if not taking a bus.

As for not flying on the sabbath- BA have to comply with the Equality Act, so make reasonable adjustment- a reasonable adjustment would be flying at a later date, but think it is unlikely paying for duty of care until then when other flights are available would count as reasonable- so you might have to foot your own hotel bill if you decline the earlier flight.