I think, perhaps, that by wishing both desktop to be available in the same OS installation, you will be asking too much of your machine. Many of the things that you like about Ubuntu Mate will be the things that, despite it lightweightness compared to the likes of Unity etc, add to its weight as compared to the kinds of desktop you would like to install for your games to run on.
Having run multiple desktops on the same Linux installation myself several times, experience has taught me that the lighter of these never runs as light as it would if it was installed alone. Xubuntu/Unity versus Xubuntu alone or Gome/Unity versus Gnome alone to cite just two examples. Thus, my suspicion is that if you were to install a second, lighter desktop on Ubuntu Mate on the expectation of it running as light as you have previously experienced it running alone, you are likely to be disappointed, especially if your hardware is somewhat limited in the first place.
Therefore, my strong recommendation would be to install them side by side as separate installations on separate partitions and then just log into one or the other from Grub depending on your needs in a given session You could even have a third partition you could use as a shared storage partition for them both.