For the breeder tank, considering it’s an aquarium, should I put netting over it as well?
You don't have to as long as there are not other animals in the house and the decor doesn't allow your dragon to jump out of the enclosure.
a glass tank is going to be too heavy to pack around all the time
The goal would be to leave the 40g breeder at the second residence instead of the 20g tank that is currently there.
If you aren't able to leave the 40g breeder in your other home so that you don't have to travel with it, you should look into a lighter weight option such as a large plastic tote.
The best lighting in the world isn't going to help an adult dragon in a 20g enclosure. In a space that small, the dragon is not going to thrive. It needs more space.
-Brandon