Native Steam client tries to launch VR games using the Flatpak version of Steam (which then obviously fails)

**Game information**
All VR games, such as "Wardens of the Amber Cage" (which is small & free to play, so it's an easy test case).
You obviously also need the "SteamVR" tool installed in the native Steam client.

**Distribution name and version where applicable**
Ubuntu LTS 22.04 and Ubuntu LTS 20.04 . (If testing with 20.04 then beware that many VR games don't work with Proton 8.0-x, so you need to change to Proton 7.0-x.)

**Flatpak info**
Flatpak 1.12.7
nvidia-535-86-05
default
host

**Problem description**
When both the native & Flatpak versions of Steam are installed, and only the native Steam client is being used, trying to run any VR games will fail, as the native Steam client tries to launch the VR game using the Flatpak version of Steam! The following symptoms are normally observed when trying to run a VR game:

1. The SteamVR tool starts, if it wasn't already running.
2. Steam seems to run a second time (and it's not obvious this is the Flatpak version, so originally I was blaming Valve)... and this typically causes the original native Steam to go off-line (e.g. on the Friends window).
3. If the Flatpak version of Steam has the VR game installed (e.g. say due to using something like "mount --bind $HOME/.steam $HOME/.var/app/com.valvesoftware.Steam/.steam"), then the VR game starts in some fashion, but only visible on the monitor (not the VR headset) - often with black unresponsive windows, or if you are lucky the non-VR version of the game.

I'd thought Valve had broken VR support on Linux, because all VR games had stopped working since I'd last tried them, and it was not obvious to my untrained eye that the Flatpak version of Steam was starting. I'd tried reinstalling all sorts of things & changing various Steam settings, without any luck. I then went to the trouble of installing a clean version of Ubuntu on a spare HD I had, and found that strangely Steam VR games worked fine. So I went back to my normal Ubuntu installation, and tried to work-out where it was going wrong (starting the second copy of Steam), when I noticed a few things which hinted that maybe Flatpak might be involved.

But as I said, this was NOT obvious to the untrained eye, _so if it's not possible to fix this for some reason, then there needs to be some kind of warning to not have the native Steam client installed at the same time as the Flatpak version_. BTW, I had both installed because I wasn't sure how compatible the Flatpak version was compared to the native client, and so I wanted to compare them.

**Does this issue reproduce with native Steam**
N/A