You can get a stronger, offsite
scooter that will have two batteries and will last better than the smaller ones. These stronger offsite scooters still won't be as strong as a Disney park scooter. Talk to the vendor you want to use and give all your details like his weight, how long you want to be in the park, and see what they recommend.
Then, you need good habits of turning your scooter off during shows, parades and fireworks. That will conserve the battery. You may have to look over yourself and see that the scooter is really off.
We just rented from an offsite vendor and I was shocked to find out that they actually tell you to top off the battery in the park. I think it's because they want to cut down on service calls. You have to find a park receptacle and so you need to look hard for one. It seems you can find one by restrooms for some reasons. There really aren't charging stations, perse. When you eat is definitely a time to charge the battery some. When doing a theater show you may find a receptacle nearby. A little charge here and there can help. I don't feel Disney CMs will be too helpful, here.
I feel the best approach to this situation is to take a break in the afternoon. Then, in your resort room you can give your scooter a good charge again. I feel this is the safest way. Then, go back in the evening for more park fun.
I think a complete charge is like 5-8 hours depending on how much you have used the scooter that day. If you have only gone down half way with your battery then it won't take as long to charge up as if it was completely low. Charge your scooter fully every night.
If you really want to be a diehard in the park you may have to do a combination of park and offsite scooter together. I think they will let you store your offsite scooter while using a park scooter. Of course, this doubles up on the cost each day.