I'm thinking your options are either the North Stratford, the Grand Union or the Brimingham and Fazeley (top, middle and Bottom roads)
The first two mean doing Hatton, the second two either Garrison or Aston then Farmers bridge. I'm assuming you aren't up for more esoteric routes like Birmingham and Fazeley, Perry Barr and Ryders Green?
The easiest is probably the North Stratford. The Grand Union/Camp Hill/Ashtead/Farmers Bridge is less used. The B&F, Aston/Garrison, Farmers Bridge is pretty standard Warwickshire ring
So - what do you fancy? My Esoteric route is quite a surprise...
Richard