Alright, here's where we're going to get into the nitty gritty. If I could make a recommendation, the first thing you want to do before you edit your profile is
disable it until you're ready to test - just uncheck the "ESO DualShock 4 Mapping" group in the Programming Items list. That can save you a lot of pain at the hands of accidental button presses.

Because of the way ControllerMate recognizes controllers that are connected to your Mac, you are almost definitely going to have to replace each of my "Wireless Controller" buttons with mappings to your controller's corresponding buttons. To do that, follow these steps for each button (I’ll use clicking the left stick as an example):
1. Use my label ("Left Stick Click") as your guide to what's what rather than the button number (I don't know if the numbers will line up between my DS4 and your chosen controller.)
2. As an example, go the "DPad and Sticks" page in the profile I provided, and look at the Button Building Block labeled "Left Stick Click", which is button #11 for me.

3. Click your left stick and see which Button Building Block lights up in the Building Blocks palette. That’s the one we want to replace mine with.
4. Add the mapping for your Button Building Block by dragging it onto the page from the Building Blocks palette. (For now you should leave mine there as a reference.)
5. Then click your left stick again - the output at the bottom of the Button Building Block you just added should change from "OFF" to "ON" whenever you click that left stick.
6. Click back and forth between my Button Building Block and yours, viewing their properties in the Inspector palette, and change all of the settings in your block so they're the same as mine - particularly under the "Modifiers" and "Behavior" tabs as I have made quite a few adjustments there to many buttons.
7. Once you're convinced they're identical, pull the "^" Single Key Building Block (that's actually the [Ctrl] key) away from my button #11 Building Block so you can see the connection line (and pretend your name is Belgarion, since I first wrote these steps up for
@Belgarion a few months back.

):

8. Grab the "OFF" from the top of the "^" Single Key Building Block and pull it away far enough that it disconnects (when you let it go it will disappear - that's ok):

9. Now grab the "OFF" from the bottom of
your Button Building Block and connect it to the top of that same "^" Single Key Building Block:

10. Now, when you click your left stick, the connecter that says "OFF" at the bottom of your Button Building Block and the two "OFF"s on the "^" Single Key Building Block should all light up and change to say "ON" for as long as you hold that left stick click down.
11. If this worked then repeat this process to replace all of my "Wireless Button" Building Blocks with your own and get your configuration all sorted out. If it didn't work, then let me know more specifically what you're doing and what is (or isn't) happening as a result.
Once you're satisfied that everything has been transferred over, you can delete my Button Building Blocks.