I don't know what resources you have to work with, but adding another or much larger air
pump you can increase the amount of dissolved oxygen, and increase water circulation (especially with a large root mass). Another thing that can help is adding some hydrogen peroxide (H2O2), about 5mL per gallon (1 Tsp). Hydrogen peroxide is purified water and oxygen, so it can help raise the dissolved oxygen levels. But it also dissipates fairly quickly, I add that amount to all my systems weekly (sometimes a little more), and with roots that are using up the oxygen so fast, I'm not sure how much help that would be. But with a larger reservoir, the the larger volume of water will be able to hang onto the oxygen molecules longer. Simply because there will be more oxygen molecules for the plant to take, so they wont be depleted as fast.
Also if the entire root mass is submerged in the
nutrient solution, you may try to lower the water level so that about one third of the root mass is above the water line. That should allow the root mass to get more air, and still not dry out. But it cant be stagnant air, so I would add some air holes at the top of the container, and with a larger air pump it should get some air circulation.
And I'm still not sure what water supply you are using for the nutrient solution, but tap, and water from a water softener may have a high salt content. So I would suggest using some type of filtered water instead.
P.S But in the long run, as you mentioned you might just want to transplant it into coco coir, and run it as a
drip system. That would also be easy to add more tomato plants to the system, and still only need one reservoir, as well as one pump to grow them.