Absolutely DO NOT limit your workset weight to accommodate DOH (double overhand) grip. Use DOH for your warmup sets, and then switch to alternating grip for worksets (and any heavier warmup sets that require it).
You can also try the hook grip, if you don't like alternating grip, but for hook grip, start using it with the lighter sets, to get your thumbs used to it.
This is all covered in the book. Re-read the deadlift chapter, it's all in there.