I find it hard to believe the OP would choose to create such a complicated lifestyle. Living in a tiny RV and NOT travelling would be torture for most. Add to that the impractical and even cruel practice of leaving a dog inside all day is crazy.
Do yourself a favour and the dog too by finding it a proper home for it then it will take the stress off you and the dog.
If you are considering buying a brand new small trailer, why not consider a used unit for the same price that would be bigger and more suitable? Have you considered that if you do not have hook ups where you stay, you will have a couple of days of water and sewer capacity then you will have to haul it somewhere to dump and refill. That will get old very fast.
