The Arctic Fox 31D would fit the bill perfectly.
Here is a
LinkIt's outside your price range but will be worth every dime...especially when your family is not shivering.
Be warned though...RV'ing in sub freezing weather is not cheap. My dad ran about 30 lbs of propane every 2 days in 15° weather keeping the fiver at 65°....that was with a supplemental electric heater.
His Challenger IS a four season RV with heated tanks, it just cost a lot to keep 30 plus gallons of water from freezing and the family comfy when it's that cold. An Arctic Fox would have helped my dad tremendously simply by their insulation.
When he works in sub freezing weather he puts the Fiver in storage and rents a pay by the month hotel or condo. It's actually cheaper that way for him. But a TT is easier to heat because of the lower ceiling height.
I am not trying to discourage you, I just wanted you to know what you are getting into.
Thanks!
Jeremiah