I dont think pullups are gonna do much for your goals (hence why ive pretty much cut them out of my program)...I think for a back movement you should be doing deads, good mornings (from pins or chains if possible), rows, power shrugs, or rack pulls (maybe some power cleans if you are good w/ them). Maybe you could throw some pullups in on your recovery day.
As far as Texas Method, I personally still don't think it has enough volume to be optimal but its still a really good program. I think its strengths are its ease of use and how its setup to give feedback on what is causing progress to stall.
I dunno thats my thoughts for now, gl
-Mike