I prefer the factory spring type clamps, they are constant tension, but the screw type should work.
For a leak to cause foaming, most times it has to be on the suction side, between the reservoir and pump. On the 3FE, the reservoir is mounted on the pump, sealed by an o-ring. If it were sucking air, would expect it to leak fluid when the motor is off.
There is no external filter on the system, the reservoir has a screen in it. Adding one, IMHO, is not recommended, they add restriction and hydraulic systems are all about flow. If your worried about contaminants, simply change/flush the system more often.
It is possible/likely for foaming to happen without a leak, the vapor bubbles are created in the pump by cavitation. This is most often caused by a restriction in the suction side and/or fluid type/viscosity. This is what your problem sounds like, especially because it is only happening when cold, when the fluid is thicker.
Toyota sees them as "not serviceable" buy a new reservoir, but, it is possible to get to the screen. Remove the reservoir, pry the tabs open that hold the top plate on, remove it, inspect/clean the screen, put the top back on and recrimp the tabs. I use a bit of RTV in the gasket. In lots of years of working on them, have only seen a couple that were dirty, clogged enough to cause a problem.
I would not use any additives in the system, they are not needed and long term likely to cause more problems than they solve. ATF is good fluid, for an auto trans.

Often it works well enough, but it's not the highest performing fluid for a purely hydraulic system. ATF has lots of additives to deal with the clutches, etc, a power steering system has none of that, so they would be contaminants in this application.
Maybe try a different fluid? I have had great results from running light hydraulic fluid (Mobil DTE light) in Toyota systems, in my experience, a higher performing fluid for the application. If you don't have a ready supply, the power steering fluid listed for earlier GM's is basically the same stuff, works well. It is true that ATF is spec as replacement fluid, but they came from the factory with hydraulic fluid.