You could try the Sleeper I have done Inverness (INV) to Euston (EUS) but this was some years ago although I understand that they have updated the product slightly The first bit of my Trip Report to Cornwall by sleeper (see my signature for a link) covers the best berths on the train carriages and reservations, but the best place to start is scotrail-caledoniansleeper-index which will allow you to check for Bargain berths which are the cheapest tickets and has other info on the service. Also have a look at The Man in Seat 61 which has photos of the train and a fuller description of the onboard service.

There would be space under the bed for your stroller and possibly your luggage although the space under there won't take thick (i.e. deep) suitcases. Would probably recommend seeing if you can get two cabins next to each other, the interconnecting door opened and spread yourselves/luggage etc between the two.

If you are on a budget be quick as bargain berths sell out very quickly and it isn't that cheap otherwise.