As has been said already, there is no self-contained distribution of Ubuntu-MATE that works with the Raspberry Pi 4. However, that does not mean you cannot end up with it. Moreover, except for one and a half "blips", it's not at all difficult:
Go to the Ubuntu web site and download your choice of the 32-bit or 64-bit Ubuntu server image for the Pi 4. Write that to the SD card as you would any other image. Boot this, log in, install MATE as your desktop (from the command line, as there is no desktop present).
Except for not having the Raspberry Pi tools, you'll now have Ubuntu+MATE on your Pi (instead of Ubuntu-MATE.
Now as for the blips I cited: You'll need to connect the Pi to your network over Ethernet for the initial boot/setup, as Wi-Fi won't work (easily anyway), and you might not have video output. That will depend on whether the monitor's resolution is that of a valid mode or close to one, but readily fixed in the manner you've already done.
There are two videos on YouTube for getting this setup going. One is by "NikoD" (or something close to that) and the other by "ETAPrime". As best as I can remember, either mentions the ethernet or the video thing, but each does add additional things the other doesn't (the combination of both gives you a more complete setup). You can ignore the "cannot see all the RAM" warning from NikoD (it's been fixed since), but DO incorporate zram anyway.
Except for the lack of those Pi tools (you can add some of them manually; there are how-to guides for it online), I've found this arrangement to be much more stable.
There are links to the videos in question and more in the following thread: Is Ubuntu Mate available for the Raspberry Pi 4 yet?