You can still use the
BRK instruction. But the reason you're not getting to BASIC is because the KERNAL is banked out.
When the 6502 executes a BRK instruction, the processor jumps to a routine specified by an address stored inside the KERNAL ROM. If you look at addresses $FFFE-$FFFF, in the KERNAL ROM you'll find that these point to a routine. When
BRK is executed, the CPU will start running that routine. So what you need to do is when the turn the KERNAL off, you need to store the address of your interrupt handler in $FFFE-$FFFF. Then when you execute
BRK your interrupt handler will be the next thing to run.
But I wouldn't really recommend this approach to return to BASIC. If you are doing raster tricks or using interrupts for anything else, it can be tricky and wasteful for the interrupt handler to try and determine if it needs to return to BASIC.
To your question:
What would you recommend for breaking either into the BASIC screen or into the monitor when ROM is not mapped?
I would recommend SHIFT+RESTORE. The RESTORE key performs an NMI, so it will invoke a routine of your choosing, (which is not your interrupt handler). That routine can see if the SHIFT key is also pressed, and if so, can re-engage BASIC and KERNAL and jump to there. Of course, you can also jump to that routine under software control.