I've experienced the rear end dancing around under braking as well. Particularly noticeable at turn 1 at Laguna Seca. 2:21 in this video seems relevant:
So the rear toes out under droop, which means it also toes out under braking. The first three potential solutions that I can think of:
A) Add additional rear toe-in, easiest and cheapest route
B) Increase front spring rate to reduce dive, which probably means revamping your entire suspension setup
C) Alter the rear anti-squat geometry with a product like the one in the video
I can't do C because it's illegal in STX and I'm not super keen upping the spring rates on my daily driver, so I've settled on option A. Factory spec is 1/6th degree of total rear toe in, I've basically doubled that to 1/3rd of a degree. It helps noticeably but doesn't solve the issue entirely, I might bump it up even more next alignment.