Hey, Helen. This is a very common question for this kind of balance and, to be frank, a major annoyance with the design. To put it bluntly, these are a huge pain to install even in the best of circumstances.
The intent of the design is for you to "feed" the balance assembly through the cutout, typically putting the shoe in first and then guiding the rest of it in after (or vice versa, whichever is easier). But good luck doing that while also keeping all the springs in place, and keeping the frame spread enough to snap it in. I can't tell you how many times I've scraped my thumb installing a coil balance.
The only trick to it is the one you've already surmised: cutting the hole so that it's a little bigger. You don't want it to be too big, though, since the last thing you want is for the assembly to fall out of the hole during normal operation. But you can afford a little more space to make things a little easier.
Otherwise, the only thing I can tell you with certainty is that you're not doing anything wrong. In fact, your situation sounds pretty dang familiar to my daily routine. That's just how these balances are, unfortunately.
If you haven't already, I recommend watching our video on the topic. We go over a few of these points and you may find it helpful to have a visual example.