Some info/musings/experiments on the AD360 batteries:
My two original AD360 batteries are now stuffed - ballooning out and distorting the case. With the knowledge that you can run them with the 12v recharge cable plugged in, I made the decision to rip the 11.1v battery out and graft in an ebay 12v LiPo unit of a size that should fit but with lower mAh rating. I could live with that.
The battery was cheap enough, and it almost but not quite fit into the enclosure. The issue was an inbuilt on/off switch on the new battery which I decided was redundant and would allow the battery to fit perfectly. So, I had a glass of stout to steady my hand and set about poking about the insides of a fully charged cheap chinese LiPo battery with a soldering iron. (Not my smartest decision)
The good news was that I managed to successfully graft the new battery into the Godox pack without burning a hole in my dining table.
The bad news was that after all that, it didn't work.
The AD360 now flashes the leds briefly before shutting down. My assumption was that 12v is just too high a voltage for the AD360, which I confirmed by removing the LiPo pack from my second dead battery and trying to power the AD360 from just the 12v charging cable - same result. (which in hindsight, was a test I should have performed first) I read somewhere once of claims that Godox do not recommend you power your strobes with the charge cable attached - this would be why, but would only be an issue if the battery were fully charged.
Plan B has me installing a DC/DC converter and a capacitor in the battery shell and powering the unit from a remote battery pack. (Looking into a way to easily connect my 18v Makita batteries)
Godox 360 strobes will run on 11.1v, but not 12v.