The circuit etch solution is made from CuCl$_2$ rather than CuCl. https://www.instructables.com/PCB-Etching-Solution-Cupric-Chloride/
After etching/dissolving some copper, you would get CuCl, and regeneration involves reoxidizing the Cu$^+$ to Cu$^{2+}$. When you use air, not all the O$_2$ is captured, and none of the N$_2$, so you have lots of gas bubbles forming and rising. One way to reduce the foaming would be to reduce the rate of aeration. A possible cause of foam is the presence of surfactant or a film-former from reaction with the circuit board.
One way to defeat the foam is to treat it with a low surface tension agent. Isopropanol (rubbing alcohol (70%) or 91%) sprayed onto the surface - just a little bit - can break the foam and let it settle down. Most of the isopropanol will evaporate easily so the treatment will have to be repeated. I have done this on a large scale.
Another additive would be a silicone lubricant. Spray cans of this type of product are used for nuts and bolts type of mechanics. A little goes a long way, and could interfere with your etch, but has potential.
Consider using hydrogen peroxide to reoxidize your Cu$^+$, as in the link above. This would significantly reduce the bubbling and might give you more control over the regeneration, although it does dilute the product.