Electrical problems are a PITA.
Have you checked for codes? tested the alt/batt?
For multiple fuses to be blowing you have to have some sort of abnormal surge. The first thing to do is to check your alternator. sometimes when alts go bad they start surging instead of just dying. If you don't have the testing equipment at your own home or know how to do it with a diode, take it to autozone and have them test the battery and alternator. This actually fits in with the stalling situation because stalling the engine also stalls the alternator...
Now if the alternator is good, It sounds like you have a short somewhere. Where is the short is hard to say. Its less likely, but you may also be having a grounding issue. Check the grounding cable just because thats an easy thing to check.
the short situation is just going to take a lot of looking... haha.
Oh yeah, and if it only happens when you stall, have you thought about not stalling? (haha sorry I had to
) - Really though, does it ONLY happen when you stall? or is the dash panel only when you stall and the BAS is happening at other times? They could be two separate issues, or they could be related. If they are related, how? BAS and Dash both involve the BAS-dash light, so that could be the source of your short? Additionally, I've only stalled once and thats after I spun out on an autocross course- I dont remember if thats what happened, but on other cars I've owned, all the dash panel lights lit up when I stalled out. So if theres a short, I'd start finding ways to engage each of the dash lights to figure out which one it is. Turn off ESP and you'll engage the BAS light for example. Most of these turn on when the car starts though which is what makes it seem a little quirky. If its the BAS system it should be throwing a code. the lighting systems likely won't.
I guess in short, check the battery and alternator. If they check out, double check the grounds and start looking for a short. check for codes and Also get back to us on when the BAS fuse flows. Does it blow when you turn ESP off? When ESP engages?
(What I'm trying to do is make sure we are isolating the BAS system from the ESP system because both share some items. If it ends up being BAS only, its likely the electric BAS pump causing the ground, a lateral movement sensor or a wheel speed sensor)
I've been a member on probably close to 50 forums and this is by far the best, most helpful forum I've ever been on, so I'm sure you'll find it very helpful. I think most of us are a little older than those on other forums, so that may be why we aren't quite as quick as others. (I am also active on the Dodge, Camaro, and fiero forums as well as vstrom, jeep, vettes, 3kgts, and Fbodys) but don't give up hope just because we are a bit slow in responding
its worth it. (especially on weekends and holidays... who wants to be on a forum when they can be spending time with friends and family?)
Hope that helps!