Disclaimer: I haven't tried it.

BUT: I'm both a Deck user and Linux desktop gamer. I believe you should be able to install Epic and GOG both (the launchers) as non-Steam apps, and then they each get installed to fresh Proton prefixes in Steam. No additional tools should be needed besides Steam itself.

I think you can still select a default install location for their games too, since you do get Linux file system access from the prefix (it's mounted as drive Z).

I'd probably run them both using Proton Experimental, since it tends to be better with other launchers. Or, a later version of Glorious Eggroll. The cases where something doesn't run on GE or Experimental (but does work with some other Proton version) should be extremely minimal, so using one of those for the whole shebang should give good results. You can freely change the Proton version for the whole launcher if something gives you trouble.

Yes there are other solutions (Heroic, Lutris), but ultimately this should work as well with fewer components. I use this approach for Ubi games that use their daft launcher.