Good question.

Hrmm. First janky thought... "dd the running system to the USB". LOL. Dear oh dear, do not do that, LOL.

Actually, first thought, before reading the post, just from the title, was "install gentoo", ... and that still works, ... if it's gentoo you want.

I wonder if you could do it from a virtual system... with whatever passthrough(?) or whatever needed to let it install to USB. So [in virtualbox or qemu or whatever] boot from whatever distro's live/installer [~ even, ventoy(?)], and install to USB, much like you would if running the live/installer on bare metal, except you're in virtual, and have access to your current daily driver OS and all your comforts there.

I noticed recently some distros do not like being installed to USB (~ is that a recent thing? Maybe I'd not been installing distros to USB from as diverse a pool of distros as I thought, and it'd always been like this.)

I feel like I'm forgetting some other obvious avenue. There may be just distro-specific methods. Like gentoo's stage3 handbook install method's already sufficiently flexible way [(or exherbo or decibellinux (oh noes, decibellinux.org is still down) or other distro that does a similar manual install)]. [edit: oh!]

• dd (lol, no!)
• gentoo (or exherbo etc ~ / distro-specific ways)
• virtual (pick me!)
• the other ways I'm overlooking [edit: debootstrap]

... Then, depending upon what you need/want it for, and need/want avoid a reboot for, other options may offer solutions not considered further back in the decision tree... distrobox, bedrocklinux, distro-agnostic package management, manual (compile&)installs, or perhaps even just using something like virtualbox gets you far enough? ... But no, none of that's what you asked for. What you asked for is better. :)

[edit: oh!: I did make a start on a distro-agnostic distro-installer script, a decade or two ago, before life interrupted. I must have thought it possible. It would have allowed you to accomplish this. ... Pity "life" happened, or I'd surely have that made already. Is doable.]