April 13th is basically impossible, as noted. Also, given that your time is so limited, you don't need two full days in Bath - if you wanted to spend that long in one place outside of London, Oxford would be much more deserving of the second day. Beautiful and interesting as Bath is, there isn't actually that much to see.
With so little time, I'd suggest basing yourself in London for most of the trip, then taking perhaps a two-night trip to Oxford. Carefully planned, that could allow you to spend a day and a half in Oxford itself, then visit somewhere in the Cotswolds on the way back to London.
Or do the same thing with Bath/Stonehenge instead of Oxford/Cotswolds.
You could also perhaps do Windsor Castle as a half-day trip out of London, if you really wanted, but I fear Salisbury might just be over-reaching here.