As for fixing the problem by changing the plastic: injection molding is an unfortunate process in that it has very high fixed cost and very low variable cost. Once you have a mold, you can churn out parts by the thousands in the blink of an eye, but getting a good mold made can easily run into 6 figures. If the gap were too wide, you could possibly grind a little bit out of the mold, but getting the gap to be wider probably cannot be accomplished without a whole new mold.
A different question is whether the cord needs to be as thick as it is. The CT Mini has a 3x12AWG with 105°C rated insulation. I don't have calipers handy, but it seems to be at least 10mm thick. Next, we need to know how much current a CT can draw. The publicly available specs are confusing. The nameplate mentions "8.3A" and "∑10A" (no idea what the ∑ stands for), the flap over the auxiliary outlet tells you to hook up at most 3.1A. The manual claims the unit uses 400–1200W (presumably depending on the hare/turtle knob), and the load connected to the outlet on the CT another 1200W (which would be 10A, way more than the 3.1A specified on that flap, which we can only assume was erroneously copied from the original 230V model of the CT). So we have a total power draw of 8.3A, or 10A, or 1200W+1200W = 2400W, which at 120V is 20A. Strictly, US electricity is specified as 120V±5%, so going with the lower bound of 114V, you could get 21.05A. This doesn't matter, though, because there is no thriving market for 21.05A or 10.53A fuses. The CT is going to presumably contain either a 20A fuse (quite probably self-resetting) or two 10A fuses for the motor and the outlet separately. I think 20A is the number to go with.
So... do you need a 3x12AWG cord for 20A? It's unclear to me. The NEC rates 12AWG at 30A with >90°C insulation and 25A at >60°C; for 14AWG the numbers are 25A and 20A respectively. Those numbers may be for single conductors, though; I don't know if they apply to multi-conductor cables. Actually, I don't know if they apply to appliance cords at all; I think the NEC may only be for the wiring of structures. My Mouser catalog has 4 densely printed pages of power cords, but only one rated for 20A, a 10' hospital grade model costing no less than $18.27, which does indeed have 3x12AWG. So I suppose that the 3x12AWG cord may actually be appropriate, and it would not surprise me if UL forced Festool to use one.
If I were saddled with this pickle, I would look at different cord form factors. There exist heavy-gauge flat power cords, with the conductors next to one another instead of in a circular arrangement. Can't find them in the Mouser catalog, but they are commonly used in extension cords for window air conditioners, like
this one on Amazon. Sourcing them in quantity may involve dealing directly with the Chinese and not be feasible for a hobbyist, but Festool should have no problem with it.
Meanwhile, a piece of 60-grit on the triangle pad of my Ryobi multitool fixed the whole problem in 20 seconds, much less time than it took me to write this comment!