A lot of good feedback and sound advice in this thread regarding towing capacity. I won't rehash that as I largely agree with the general concurrence.
I will instead comment on the experience of living in a 30 ft or less trailer as a teenager with an older sister and 2 parents plus 2 dogs. In short, I highly do not recommend it. It was a miserable experience for 4 people and 2 dogs long term. Not enough space, definitely not enough privacy, and not enough infrastructure to support long term living in a >30ft trailer. This was in 3 different trailers of different layouts/configurations with and without sliders. The longest trailer was 33' if I recall, all bumper/hitch pull.
Shower/bathroom was super crowded, even when empty due to all the personal hygiene needs stored there. Food storage was always an issue, even when doing minimalist shopping each day. Privacy was near non-existent and schedules had to be made in order to be away from the trailer for certain "activities" by parents. No matter how discrete they tried to be, it was always painfully obvious. Family activities were extremely challenging and always required at least 1 person to be uncomfortable, this was done via rotation but still put a damper on activities. I have a long list of other negatives but I think what I posted above is sufficient unless you want to know more.
for 4 kids, 2 adults, and 2 dogs, a 5th wheel of comparable length to a 30 ft travel trailer will give you a lot more space to rectify the above major challenges that made it a miserable experience and that's just living. A pick up with extended passenger quarters would be advisable for a family of your size with the added ability of a 5th wheel. 6 door trucks are more common than people realize. This approach probably widens the necessary budget significantly. Living in a RV with a full sized family is a lot more expensive and a more challenging to do so and do it well enough that the family is not miserable. Whatever you decide, good luck and keep us posted!
