You definitely want enough tongue weight on the travel trailer while the extra trailer is hooked up. You also need to be sure the extra trailer has enough tongue weight as well, if it has a light tongue then it could start to sway and cause the larger travel trailer to sway as well.
The ATV needs to be positioned on the small trailer so you have 10-15% tongue weight on the small trailer. Hook up the small trailer to the travel trailer and you need 10-15% (preferably 13-15%) of the travel trailers weight on the tongue of the travel trailer. To figure all this out, you will need to know the weights of the travel trailer (loaded for camping) and also the weight of the small trailer loaded.
To increase the tongue weight of a trailer, move more weight towards the front of the trailer. You could also add water to your fresh water tank if it is in front of your trailer axles. Or remove water if it is behind your trailer axles.