I suspect the answer is 'no', but I wanted to rule out the possibility that there is a way and I'm just missing some clever syntax.
The problem boils down to
outer$ bash
inner$ trap 'echo TSTP received' TSTP
inner$ trap 'echo CONT received' CONT
inner$ suspend
[1]+ Stopped bash
outer$ %1
bash
inner$ ah well shucks
bash: ah: command not found
I can already register a trap for EXIT in the inner shell with no difficulties. Being able to run short commands when receiving the TSTP and CONT signals would be such an elegant solution to the new issues that have come up, but of course stop/resume handling in job control is significantly different than simply running a cleanup handler when exiting...
Can trap be made to work in this way?