How frustrating. I would not bother with the background of the ticketing. Interesting, but not relevant for the lost bag.

the absence of a pram is immediately an issue. If you have insurance, I would contact them first, at least have it documented. And maybe they can give immediate guidance and approval or denial. BA no point doing anything until you get back.

Check to see if it is possible to rent one - I know (,) but it is the first question, I have found BA to ask. Document that it was not possible. Then work on obtaining what you need. I would stick to the minimum that works.

it is a grey area, as it technically not essential clothes, and they like the rental idea. So can take some pushing. ;-)
and just like clothing, they will not pay for a bag…

KF