Thanks for your response. I have actually owned the game for a long time, maybe even before release, I can't remember. In any case I was an early supporter so a refund at this point would be silly!
In my latest attempt, I was able to get the cursor into the region needed to press Play and hit it to get into the game. However if the cursor leaves that whole window (say, by going to the area on the left), it becomes a 'Steam Big Picture' cursor rather than the game's small black-and-white cursor, and then at that point it may get locked out of the window region on the right. So it might be a SteamOS issue, I'm not sure.
A workaround that shouldn't require any real effort, and would actually unilaterally improve the experience for Steam Machine owners (since they are after the 'console experience'), would be to have two options for launching the game. The first option would launch straight into the game using the current settings, skipping the launcher. The second option would load the launcher, allowing you to load mods and change the resolution, buggy though it may be, but at least the buggy part would be optional that way.
I understand what you are saying about not supporting Steam Machines, but I don't think it would be a good business decision. So far, your Linux desktop sales have no doubt been larger than your Steam Machine sales since Steam Machines were only just released last month. But Linux desktop market share isn't going anywhere anytime soon; Steam Machines, however, have a lot more potential. If PC game developers like you work together to create a console-like experience utilizing what's now available in the PC gaming ecosystem (i.e., Steam Hardware + SteamOS), you will be collectively opening yourselves up to the console market without having to deal with the financial and technical burden of creating separate Xbox/PlayStation ports. If you do end up making a Xenonauts 2 I hope you will keep this in mind!