My buttons arrived from LizardLick earlier in the week and I finally had a chance to swap them out last night.
I bought
this filing set from Lowes, which did the trick. Thankfully, most of the holes required very little filing.
So I went from this:

To this:

And on the top side:

Since that task really was sooo simple, I want to try doing more now. In addition to replacing other buttons on the control deck, I was thinking of going with a square gate on the joysticks. I ultimately decided that a square isn't going to make playing non-fighting games very much fun.
So I was looking into replacing the bat tops with ball tops, but it doesn't look like the joysticks I have (Happ Competition) can be replaced. I say this because LizardLick nor HappControls seem to carry ball top replacements for Happ joysticks. Can someone confirm this assumption?
Thanks again for all the advice!