Use your double over hand grip for Pulling as heavy as possible then switch to hook grip. As mentioned tape will help but I never used it, you will eventually accept your thumbs are not going to tear off and just block out the pain.
Holds for time will also prove effective.
Doing them in a rack would be advisable adjust the pins so 1) You don't have to do a deadlift to train your grip 2) If your grip fails it doesn't have far to drop. I rotate intensity/weight, shorter rest periods also heavier weights for less hold time.
Times are 60sec for less weight at which point your grip should be being pulled open by the weight to 30sec with more weight going to the same point of grip starting to fail.I always trained them for example heavy 30 sec holds for several sets then next training session switch it up to lighter for longer holds.
These should leave your forearms pumps and trashed, not sure what to recommend for sets as that will depend on the individual.
Pick up a wrist roller that you can slide over a pin in the power rack, put it at shoulder height and roll away till fatigue sets in.
As mentioned not using straps for your back training on all your pulling motions will also help but don't let your grip strength limit your training find a balance while you increase your crush strength.
Not sure if you said alternate grip isn't an option or you find little benefit employing it.
Don't like to use that grip anymore bc of all the claims of bicep injury. Having said that I can do significantly heavier pull work using alternate grip.
Picked up a plate load pinch grip design machine. You can score one of these for $120-200+ absolutely love this thing and has taken my grip strength to another level.