Okay, been attempting a few day-1 DIY fixes, pretty happy with the results I have so far. Honestly, out the box, I feel a good deal better about the PS4 controller than the Wii one.
Video of it in action to get a feel for force required/amount of sound. All these hits are done straight-armed and relatively lightly.
So, here's what I've done:
Blanket under Tatacon: Helps to reduce sound, helps keep Tatacon in place (which isn't really an issue for me because I have...)
Ratchet strap: Pretty much guarantees the Tatacon won't be budging at all
Washcloth affixed to the front via rubber band: Reduces sound by a sizeable margin. Probably a bit of tradeoff for sensitivity, but it still generally works pretty well and I'd happily take that if it means I feel more at liberty to hit it harder without disturbing neighbors.
MyBachi: Probably one of the biggest differentiators, they're significantly easier to handle and more effective than the stock plastic ones. You can actually nab a pair for under $10 with Amazon Prime if you search for "MaiBachi" (at least in the US, can't comment on other countries)
Once I gather some materials I'll try to cover modding in the future. This is all a bit overkill, but hopefully there's one or two things you all have laying around to help things out some. I definitely know the disappointment of realizing the Tatacon isn't as good as I'd hoped, but I hope some of the advice and tricks I've picked up can prove to be of use.