I know exactly how you feel, the same, exact thing happened to me. About ten years ago I was raking the leaves in my front yard when I looked down and saw what looked like a leaf attached to my leg. Well, it turned out that’s what it was. But it had some kinda sticky stuff on it and it was kinda hard to get off and everything. Anyway, I got it off and it wasn’t rabid, but you can imagine I was a bit concerned for a couple of minutes there.
NOT SAFE!
Never, ever touch a bat, alive or dead, and if you do, see a doctor immediately. Even *proximity* to a bat is regarded as extremely dangerous.
Bats are wonderful creatures, consuming an enormous amount of insects, and they are to be encouraged. But under normal circumstances, you should never be in contact with one, nor will a healthy bat willingly get in contact with you.
Dog or cat, if they are exposed to a bat they must be quarantined, even if vaccinated. If they have bitten or carried the bat in their mouth, you need to limit contact with them as well.
No matter what all else, you will probably have to get a rabies shot, so resign yourself to that.