It probably depends on how much paddling you do! I, personally, would have told BA I wanted to take the offer of the morning's flight to ensure that part was clear and protected - even if it was a note on the CSD/CSM's iPad, but certainly by phone once in the hotel if possible. Then you definitely haven't offloaded yourself or no-showed the next day. I would then have taken a view on how long sorting a hotel BA was likely to take, and if necessary taken that part into my own hands. I'm fairly confident your Duty of Care expenses would have been honoured without question under those circumstances.

Now you probably have a bit of a messy situation which should get resolved in your favour via the CEDR route, but it's not great all round because of the hassle and delay involved.