Thanks all for your comments, I haven't posted because I've been stuck in analysis paralysis about this, due to the lack of a clear winner.
My current thoughts:
iPhone
I already own it, so that's a massive advantage, and it'll get upgraded "for free" over time because I'll do that anyway. It's not small, but it's so common now that it doesn't stand out. I'm wondering if filming vertically might attract less attention and I can just crop in, which is fine considering the vintage aesthetic I'm thinking of.
GF3
I already own it and I have this completely baffling soft spot for it. I have so many other options but keep coming back to it, and if I got something else then why would I expect it to have an effect no other camera has had before it?
Panasonic ZS20
I just remembered I own this camera, and while it's not in the 4K club, it's still got lots of the cool features of the other fixed lens cameras mentioned (e.g. zoom resume) so it might be worth pursuing. The OIS was crazy good from memory too.
SJ4000 (modified with zoom lens)
I already own this and while it's only got mediocre 1080p, it is radically smaller than the GF3 and quite flexible to shoot with now that it has the ~20-80mm FOV zoom lens on it. The zoom does stick out the front and is almost as large as the camera itself, and in some ways I regret replacing the previous tiny fixed lens I had for it which was about a 50-60mm FOV equivalent. The lens ergonomics are awful and it's as far from parfocal as you could imagine and with a tiny focus throw the MF experience on the tiny screen is something that has to be experienced to be believed.
Maybe I should buy another one and install that previous lens?
For those curious, there's a thread about this camera: https://www.eoshd.com/comments/topic/35801-the-d-mount-project/?do=findComment&comment=415574
Sony X3000
I already own it and its tiny, plus the 4K and DR are impressive so I could crop into it in post significantly. It doesn't motivate me in any way though. It's strange, but I feel zero impulse to pull it out of the drawer and pick it up.
Zcam E1
Not easily available and sounds like it might not be that great a filming experience, despite it being so small and having a comparatively good image for the size.
GX850
Great camera, but very similar to my GX85, so I don't think would bring a lot extra to the table.
Fixed lens models - LX10 / ZV-1 / RX100 / etc
Look like great options but make me question why I would spend this much on something that overlaps with my existing cameras so much and doesn't really nail the vision I had for this specific role.
RX0
Size is killer and lens / image seems almost perfect, but there are flaws and questions aplenty, the price being one of them.
So, TLDR; I have so many options already that are sort-of suitable that none of the options hit the nail on the head enough to make me feel like buying anything. I might have too many nearby options that perhaps no camera would be able to get me to buy something, and even if there was one, It would probably be so niche that no manufacturer would ever even think of making it.