ABRP only works for me with Android Auto if I start ABRP on my smartphone first. If not, then Android Auto will either stop immediately after plugging in (if ABRP was active the last time I used Android Auto) or if I start ABRP via Android Auto.
This does seem similar to an existing issue, you may try this workaround;
Long-click the ABRP icon, click app info, and scroll down to advanced settings. There is a toggle button called "overlay on other apps" or similar. Please enable that and check if that improves the situation for you.
Could you tell us more about your setup? What phone model do you have and what version of Android is it running?