Put on dark clothing and a mask,and pretend you’re planting explosives in the arch-criminals lair!
You should hum one of the 007 movie theme songs (From Russia With Love comes to mind) to yourself while you’re doing it. Sneak out there with style! Get your wife to participate, too!
Making it fun makes all the difference when doing a chore like dumping forbidden crap in the public dumpster.
You may have thought of this. Do you have a shop vac you could suck up dust while you saw?
No Shop Vac, but I did try something with a paper towel tube, some duct tape, and the regular house vacuum. That lasted about 15 seconds until the bag clogged. A fresh bag lasted another 20 seconds. Household vacuums just can’t cut it.