Agree with MB. Convenience spray bottles are great until they fail.
Curious though why you need to clean your hands after. Do you not need protection for your hands? I assume they are exposed. Maybe not.
Also you may consider by spraying you are probably using much more than you need to carry. I find by wearing the right clothing bites are minimized to exposed skin. A single drop or two on my exposed hands and I'm good to go for hours. I have a mini eye drop bottle from the dollar store when filled could probably keep us going for a week at 4-5 grams. We donned our head nets this weekend and as much of a PITA they are to see thru sometimes, they are GREAT in keeping the annoying buggers from buzzing your face and ears, zero bites and less deet stink and exposure. YMMV
If you still want to dilute it, do it with regular deet liquid until it goes through your spray bottle.
jimmyb