Yeh, you're probably correct but if the mag is also not seating high enough then it becomes an issue that only shimming/getting a new mag catch can do. I went through all of this with my G&P WOC when I was fitting the new GHK mags. Bit of a farce, and initially I just sanded down and shimmed the mag well where required. Ultimately it look a new catch. Again though, as you say, the catch was less obtrusive as it is in an M14.
As you say, with the M14 the catch is to the rear so that makes me think this thing is at the wrong angle rather than being too far forward or backward which a shim isn't going to help with.
Reece, can you try and get a more accurate idea of how the magazine isn't seated correctly? Too far forward/backwards or whether it needs pushing up and forwards for example? From the video it actually looks like you're pushing it forward and upwards (i.e. rotating it), which a shim isn't going to help much with given the way that M14 magazines insert.
Edit
...unless you're very careful about it. I'd recommend perhaps shimming with epoxy than sanding it so that it's flush at the very bottom of the mag well and not at the top (basically a slight incline/wedge). That way it won't obstruct reloads. You're looking at a fair bit of work though.