I've made a few of these, and it seems like the fit can be different from kit to kit. But it's still one of my favorite wedding pens.
I used my own laser engraver, a NEJE 7w.
At the time, 7w was all I could afford. If I bought one now, I'd get a more powerful one. You can always turn it down. Wood is easier to engrave than acrylic. You have to fill acrylic engraving with something so it will show up.
I've turned a few kits where the bands or end pieces have raised bands themselves. Often, the bushings are not quite sized right for a proper fit of blank to pen, as the bushing size doesn't account for the fact that the kit parts have little raised bits themselves, and are just sized for the maximum diameter of each kit part.
