i use vigoro relaxeri know it's a very very old relaxer but it gives amazing results. it doesn't burn no matter how long you leave it in and i am almost six months post and i dont know how your new growth is but after six months my hair is still on point. and mind you i only get it pressed once every month.
i also have bleach in my hair. it's growing out but it's not breaking or anything. i use to get a relaxer and then a color rinse like three weeks later and everything was still fine. i dont know if vigoro is lye or not but they sale it at texasbeauty.com if you wanna check it out.