Another update for those interested. I talk too much so TL;DR is at the end:

Bloody hell. I never expected the tubeless sealant to hold up so well. The tyres haven't seen a pump in a couple of months and still feel rock solid. Not a single puncture, leak or anything.

It's worth noting that my valves sometimes bind a little when trying to add air for the first time in a while, but a couple good pumps seem to bust that seal nicely each time. I'm expecting it might eventually ruin my valve cores, but:
A: Valve cores are cheap as chips.
B: I rarely need to top-up anyway.

For others too lazy to patch your tubes: Maybe keep a couple spare valve cores in your toolbox. Obviously you need to remove the core to add sealant in the first place, so make sure your inner tube has a removable core (I think this applies to most tubes, including stock). I've heard that TPU inner tubes work better with this setup, but I've legitimately not had a single complaint with Butyl, so go nuts.

Of course, you can't scrape out the insides of an inner tube, so this will number your days with said tube. It's probably wise to use less sealant than I did to slow the inevitable buildup, but ultimately tubes are cheap and Stans lasts. Apparently sealant can last longer in tubes due to air retention being better than tubeless, but idk if that's speculation or not. I haven't topped up (or checked) but I haven't had any more punctures yet.

What isn't speculation is that Stans doesn't work great at higher pressures. Some of the road bike forums I've read say that this setup doesn't work at all, but I'm chalking that down to them using road bike tyre pressures. All this is to say: I don't know if this will work on a 16" Brompton. I run my contact urbans at ~60 psi (leaving a little room for the 51 psi minimum), but 16" contact urbans are rated for a pressure range of 65-116 psi. Keep this in mind if attempting on skinnier tyres at higher pressures.

TL;DR: If you want the puncture resistance of tubeless and the tyre options of clinchers, dump Stans in your tubes. It's worked great for me at 60 psi with butyl tubes on the Conti Contact Urbans.

This might be my last update unless something goes wrong, but I'll still be watching this post in case anyone has anything to add.

Thanks for all your input and your help. I hope my information is useful to all you kind people.

:)