You should be reversing the supinated hand every set. That will spread the bicep stress across both arms.
You may also be bending your elbows during the lift. Try taking a video of yourself to check. The arms should always be dead straight. Always.
Don't ever do any arm/bicep work prior to deadlifts.
Yes, you can use straps. Its better to use straps and healthy untorn biceps. Surprisingly, they don't affect grip strength that much as you still tend to grip like hell, even with the straps doing the "heavy lifting". You can also do separate grip training if you feel that it's a weakness.
Do bicep work as well, just not on deadlift day. Worked biceps are health biceps.
I hear you on the hook grip. Standard decent quality bars are 29mm diameter. Weightlifting bars are 27-28mm. The women's bar is 25mm. The reason for that is the hook grip, giving a touch more room. Personally, I think it's too painful for deadlifting. You should be focused on pulling the weight, not focused on pain in your thumbs.