Dear rkauzlarich: you can do it! You might think about arranging your air in to London (for the GCT tour) and home from Liverpool. Book a hotel near Albert Dock for 3 nights in Liverpool and make your plans.
Day 1- morning train Edinborough to Liverpool - this is a 4 hour ride. Trains are fast, clean and run on time
on arrival visit the Beatles Museum on Albert Dock (www.beatlesstory.com)
Day 2 - do the full day Beatles Tour (http://www.beatlestours.co.uk/) -- you should book ahead
Day 3 - see if the National Trust is offering tours of the Beatles childhood homes (http://www.nationaltrust.org.uk/beatles-childhood-homes/) Book ahead
Fly home ( from nearby Manchester airport, closest international airport) the next morning.
All this would entail is a "breakaway" with GCT, if you are taking their air; or just go ahead and book your own air. Good luck with this -- I've done this exact thing ( train from London) and it is very doable.