I think I found another way that can be done without using the desktop mode: In config/config.vdf you can add this to the bottom of the Software/Valve/Steam section:

				"Tools"
				{
					"enable"		"1"
				}
				"ToolMapping"
				{
					"0"
					{
						"name"		"proton_37"
						"config"		""
						"Priority"		"75"
					}
				}

Could be totally wrong, but it was the only mention of proton I saw when enabling this on my desktop.

I can confirm that editing the config.vdf directly and adding these lines works for me.

I still prefer @mtdeguzis 's solution as it's much easier to select proton versions in the gui instead of remembering the version name as a string. However I needed to type
steam -nobigpicture
in order to change the settings, otherwise it would go straight to BPM and then trying to exit that would cause steam to hang.

anyway these is an important feature request for steamos.

Dude!!! can't watch to try this out the. Thanks!

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From: kisak-valve [email protected]
Sent: Wednesday, March 6, 2019 6:50:12 PM
To: ValveSoftware/steam-for-linux
Cc: Deguzis, Michael T.; Mention
Subject: [External] Re: [ValveSoftware/steam-for-linux] SteamOS - no settings option to toggle "try with all games" (#5631)

Global and per-game Proton settings in Big Picture Mode implemented in the 2019-03-06https://nam05.safelinks.protection.outlook.com/?url=https%3A%2F%2Fsteamcommunity.com%2Fgroups%2FSteamClientBeta%2Fannouncements%2Fdetail%2F1798531141142819058&data=02|01|mtdeguzis%40geisinger.edu|05519712d7ba4dce3a3408d6a28e79d4|37d46c567c664402a16055c2313b910d|0|0|636875130161470105&sdata=4taptWm6LPjouWxpEcgAFY2LQfRYHOlfiorxZjsfAR8%3D&reserved=0 Steam client beta update.

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I can confirm the new UI is in BPM and appears to function properly