What the OP actually needs is a bike. Preferably a push bike. A motorbike needs more care with having to put in fuel and oil. We can all see that is too much for her to want to do.
A push bike if you only went to see it a couple of times a year would virtually be the same. No loss of training. If kept in a shed maybe wouldn’t rust.
Move this horse to their own property and move it far away from the dogs, means the horse will be neglected. No worming, no teeth done yearly, no hooves being done every 6 - 8 weeks, no picking hooves out daily, no putting on a day rug and a night rug on in winter, no daily grooming, no feeding 3 times a day, as there was grass there when checked when put in paddock, no need to check water, or as there was water in a dam when put in. No need to pick up manure, as that doesn’t happen in the wild. No vaccinations done. Fences don’t need to be fixed as they were okay when put in.
The other horse won’t be able to be ridden as no training kept up, so can’t catch as it is out with the original horse, and if it does get caught will have separation anxiety.
Horse/s will die or probably end up killing the OP.