Boss Rush is not a good training partner. I beat it by throwing all of the enemies (with the exception of Dust and gAkiha, which can be AI exploited in other ways) repeatedly. It falls the the same AI abuse as normal play.
If you want the best practice without a partner to play, watch high level videos and use ALL of the tools available in training mode to emulate the things you watch.
1. Practice combos. Reliable execution is an easy way to step up your game.
2. Set the dummy to Crouch, Guard All, and Recover A. Practice staggers in blockstrings to beat mashing.
3. Set the dummy to Crouch, Guard All. Practice longer blockstrings including blockstring resets and mixups within blockstrings.
4. Record blockstrings that you see other players do for the Dummy, and practice how to defend and escape them.
5. Practice okizeme mixups, both after a BnB and after a random far range sweep.
6. Record the dummy to do wakeup reversal or mash 2A or shield. Now after you sweep them, you can try to find out different methods of baiting it out (as if you were going to do oki mixups) and punishing.
7. Learn about option selects, how to use them, and what to use them against. Use Dummy mode to practice.
When watching videos, the most important thing is to try and understand WHY the high level players do what they do. Pay attention to everything: their movement patterns, when they use each normal, how they get out of pressure, how they make use of meter, etc.