Good question.
I use a Hogster-r 35mm as my scanner, and I use rechargeable batteries for it. There's a few reasons I wouldn't use them for my rifle scope.
First, even though Bering Optics says that you can use rechargeable batteries, they don't really recommend them.
Second, my Hogster-r eats through rechargeable batteries like a fat man at a buffet. The one minor irritation with the SH is the almost non stop auto nuc when it is first started. Running a battery pack and leaving the scope on all night helps reduce the auto nuc.
Third, while there is some added weight, it's not noticeable enough to not use it when you weight the pros and cons.