Hi corbin,
I'm always happy to help! Tesla coils have been a hobby of mine for nearly a decade, and an interest for even longer. I have been blogging the build of my dual resonant solid state Tesla coil here on ETO, though it's been on the back burner for a while now due to certain life events. I have been learning as I go. Tesla coils are a fascinating subject!
GDT toroid material is critical, as is the length of the secondaries. Do you have the ability to scope the output from the GDT? The following link is very thorough and helpful when designing a good GDT:
https://www.richieburnett.co.uk/temp/gdt/gdt2.html
It explains what can go wrong with GDTs and how to fix it. As I mentioned, GDT core material is critical. It must have a very high AL value (provided in the datasheet), preferably of 10,000 or higher. They are much better at higher frequencies and are more reliable at switching the transistors. It is also very important to keep secondary lengths identical, otherwise the extra inductance will cause delays in the switching signals, which can cause your H-bridge to switch unevenly. This could cause damage to your transistors. It is also very important to wrap the primary and secondary conductors very tightly around the toroid to reduce leakage inductance (which can also cause switching delays). A great way to build a GDT is to use CAT5 cable (leave the conductors in the sheath) and wrap it tightly around the core. This ensures that the conductors are identical in length.
In case you're interested, here is the blog for my DRSSTC:
https://www.electro-tech-online.com/blogs/building-a-dual-resonant-solid-state-tesla-coil.248264/
Keep in mind that I was learning as I went, so lots of things change throughout the design/build process.
I expect your bridge is staying cool because you are running it at such a low duty cycle (a very good idea). It gives the transistors time to cool down between bursts.
Regards,
Matt