If you were to commit to 4 years of RV life, you should be able to get a better lease rate than 500/mth. Check into renting a park model or leasing at least for a year time frame.
I spent a year living between a 20ft travel trailer and a 27 ft sailboat with cabin. Was newly married and teaching. It worked but after a year I realized we could not go on indefinitely.
I will say that your heat situation can be helped by using a reliable electric heater with thermostat. Having 2 dogs, have learned that it is safest to position tabletop. This will save a lot on propane heating esp if your elec is covered in rent.
Agree that 30 ft would be a good length. If the park has good bathrms, you can use their shower facilities to free up your camper shower space for storage such as hanging clothes rod, litter box, etc. When choosing camper, consider the space for clothing. There are many that offer only a small wardrobe on sides of bed.
Also, consider that if you are gone several hours a day, your dogs will have very limited space for movement in camper vs a small apt.
I'm not convinced a camper is your best move but if you both have a "rustic" personality, you might pull it off.
If you are a handyman type, you have better odds of succeeding.