I think you've got the right idea. If I were you (other than switching gyms or buying my own equipment) I would use straps for working sets and supplement with grip training. I pull with a hook grip and use straps probably 90% of the time in training but have never had a problem gripping the bar in a competition.
You can try some variations that will still allow you to build strength while using less weight:
Pause deadlift (off the floor, below the knee or above the knee)
Stiff leg or RDL
Or Any combination of the above.
If you pick a few of these variations and get stronger at them, your 1RM competition style deadlift will most likely go up as well. After your working sets, you could throw in some static holds, either double overhand or mixed grip (or both) for your grip.