Trump is an isolationist, and he announced his goal last week to pull troops out of Syria, so I don't think Trump has any intention of leaving US troops in Syria. Bush talked about regime change in Syria. Obama talked about regime change in Syria. But I don't recall Trump saying anything about regime change. Trump apparently is ok with leaving Syria in the hands of Russia.
As you point out in your post, we are in a classic lose/lose situation.
IMHO Trump's now telegraphed missile strike has nothing to do with getting more deeply involved in the Syrian civil war. I think Trump is making a statement to other audiences as well.
- Syria made an agreement with the UN to destroy their chemical weapon stockpile. Obama did nothing when Syria used chemical weapons their own people. As the anti-Obama, Trump wants to show that he is stronger than Obama.
- Russia was also involved in the chemical weapons agreement between Syria and the UN. This is a signal to Russia that if they want the US to leave their puppet Assad alone, then Russia needs to make sure that he doesn't use chemical weapons. As you point out, I guess it's ok to kill your own people, just don't use chemical weapons to do it.
- I think this is also a signal to North Korea. I think Trump is flexing some US muscle before his negotiations with North Korea in an attempt to show what the US can and will do in response to the use of wmd.
- Let's not forget Mueller. I think part of this involves Trump trying to show that he can stand up to the Russians in an attempt to show that he is not "colluding with the Russians".
- And finally, as K points out, this is mid-term election year, and Trump is trying to increase his popularity and avoid losing one or both houses of Congress by rallying the country around the flag as he stands up to the evil Assad and Putin.
So I really don't think the impending US missile strike will be anything more than a punch in the face to Assad before the US packs up and leaves Syria. So the smart thing for Putin to do is let Trump have his glorious exit from Syria without doing anything that will persuade Trump that he needs to change his mind and get more involved.
My .02