Hi crackers,
I would try your mattress with the bedframe and slats you currently have first to see how it works for you. You can also try your mattress on the floor to see how a non flexing support system (the floor) compares to the one you have.
If the flexible slats don’t work well with your mattress or if your mattress works better on the floor (which would indicate that it would work better on a solid non flexing support system) then you would need to figure out a way to remove the slats and the caps (which would probably also mean removing the center support beam and any side rails that the end caps are attached to) and replacing them with a new center support beam with support to the floor and new side rails that you can use to attach solid non flexing wooden slats (typically 1x3’s with less than 3" spacing) on top of them.
You could also try the plywood as well although as you mentioned it may not be completely stable or may make some noise if it moves or shifts on the flexible slats but your own experience will let you assess this. It would also remove any airflow under the mattress which can also be one of several risk factors that in some cases can add to the risk of mold and mildew under the mattress (see post #10 here) although if there are no other risk factors involved it would probably be fine.
Phoenix