Lower back exercises to strengthen?
Bent over barbell and dumb rows work well, lower back extentions as well as dead lifts. Also squats with a bit of weight on your shoulders. Look at core exercises as well like mason twists or knee/leg lifts.
Straight leg dead lifts. Just make sure to keep the weight lower, anx remember to always keep your back straight.
I think it will keep hurting whenever you bend over for long period of time. In my experience strenghtening is a part of the solution but it will never fix it completely. Bending over for a long time puts pressure on your lower back, whatever you train that pressure will be put on those muscles. So keeping a good lowerback and spine position is also really important in not making your muscles cramp and contract constantly.
Deadlifts are for me the most superiour exercise for strenghtening the lower back.
I dont see any problem in you doing lower back training two times a week. Just dont go all out two times a week just on lower back. That would be an overkill.