I seem to recall there was a model of Britannic in the museum but perhaps too small to be a builders' model. The triple bell steam whistle from her after funnel is there - but you probably know this already.
I can't see any structural relationship between a Georgic/Britannic model and that of Olympic/Titanic. The story goes that the one presently purporting to be Titanic was roughed up during the filming of 'Raise the Titanic' and needed much repair.
Allegedly it was originally Olympic, then opportunely changed to Titanic. It was eventually resurrected from the basement of the White Star Building to where it had been ignominiously consigned after the 1912 disaster.
It would seem that the 1/48 model at Liverpool started life as Olympic at the development stage of the prototype, was modified as changes to the design took place, and altered to Britannic later. After the loss of Britannic it was altered again but effectively became a composite of all three ships as it retained detail from all. The LMM restored the model as Titanic but it is not accurate!
Now, I seem to have seen a large model of the motorships somewhere but cannot remember where. I think that it may have been the Georgic rather than Britannic - Liverpool Maritime Museum perhaps? Can anyone confirm?