I think you need to revisit the Motorhome idea and plan to tow a small car. A TT will not have enough storage or insulation to meet your needs.
All types of RV are not created equal. TT's are nice and provide the most bang for the buck. But they are generally built with weekenders in mind. They do not have adequate storage for a family of 4 for full time use. ESpecially if you are going to be in cold weather, which will be a huge challenge in itself.
There are fivers to meet your needs, But I agree 2 kids and 2 dogs present a travel challenge. I have 2 kids and 1 dog and we have zero extra space.
YOu can find a Class A to meet your needs. Travel is a breeze, storage is acceptable. They can be well insulated and are capable of towing a small car.
A class A will be your most expensive option,however it is the only option that truly meets are of your needs.