Non-bite exposures to rabies are very rare.
Off to a good start!
Transmission of rabies virus usually begins when infected saliva of a host is passed to an uninfected animal. The most common mode of rabies virus transmission is through the bite and virus-containing saliva of an infected host.
Well, sticks don't have saliva, and they generally don't bite. So, that's good news. I haven't really seen a bat drool all over a stick or spit ever, either. So, that's a plus.
Scratches, abrasions, open wounds, or mucous membranes contaminated with saliva or other potentially infectious material (such as brain tissue) from a rabid animal constitute non-bite exposures.
Again, bat saliva? No. Also don't think the bats would rub their brain tissue all over the stick either. Poo or pee is much more likely for a bat to get on a stick, so let's move on and consider if that is possible.
Other contact, such as petting a rabid animal or contact with the blood, urine or feces of a rabid animal, does not constitute an exposure.
So there you have it, from the CDC - poo and pee don't count as an exposure AT ALL. Now, let's say a bat did somehow get saliva on the stick (which honestly isn't even something we should even consider because it doesn't happen, but for the hypotheticals and peace of mind...)
SURVIVAL OUTSIDE HOST: This virus does not survive well outside its host as it is susceptible to sunlight and desiccation.
So, exposure to sunlight and air kill it? I'd think a stick on the ground gets A LOT of that.
Typically speaking, the rabies virus will be rendered inactive once outside of the host body (and host body material such as saliva) for a matter of seconds.
Great! So, did you see a bat spit on the stick and then jab yourself with it? If not, then you're good to go.
Congrats, you don't have rabies! Rest easy and enjoy your life.