Sorry, I am not FLESHY but I think I can be of some help. I think the chainlink is the main concern here. The puffer is small enough he won't put a huge pressure on your system.
The chainlink however, obviously creates the most waste and has the need for the best filtration you can get on the tank. The size of the tank is not a problem so much as the ability to keep the water quality high with that much fish mass inside.
What are the current water parameters? particularly nitrate?
It will be absolutely necessary to maintain a frequent water change schedule with or without adding the puffer. It will be tough to keep the proper levels with him in there. I would also scale back the feeding, eels really only need to be fed every couple days. This will do two things...
1. Create less waste in the water, both eaten by the eel and remnants from the eating.
2. Slow the pace at which the eel grows. How big is he now? The smaller he stays the less waste he will produce.
Both of these will help your ability to keep high water quality.
In your situation, I would put the puffer in after removing the Hawk. An upgrade will have to be in your future either way if you are to keep the eel. This upgrade would be in tank size and filtration quality to meet its demands. So in the future, trading him in when he grows or getting a better tank will be your choice. How urgent will depend on the size he currently is?
