Well, the BPC856 datasheet says it's rated for 80V, so if it's running at 350V then it's not suitable for this design.
I'd suggest looking for another PNP that can handle ~400V. Digikey has a 400V device that has an SOT-23 footprint (ZXTP08400BFFTA), so that's worth looking into.
I doubt there's anything wrong with the pre-drivers, but you may want to poke around with your scope to make sure none of those devices get stressed more than 40V across their terminals, particularly collector-emitter. If they are stressed beyond that, then they should be replaced with a more robust device.
Since your device was off during the hot weather, it wouldn't be affected. Storage temp ratings are often 150 C for solid-state devices, so they are about the last thing that would get cooked. The case would melt first.