I get what you're trying to do, but for me, personally, it's a steeper hill than I'd wanna climb. I'd be more inclined to use an electronic flash (obviously firing with a fully open fp shutter) which means you need to have the camera on a stand. And obviously if it's on a stand then you could use your other lenses, focusing on the gg. So those lenses could sync with any electronic flash.
If I were in your shoes, specifically wanting to shoot handheld, focusing via rangefinder, and use flash with the fp shutter, I would take a quick look at some of the cheaper, newer hot shoe flashes. Not well known is that many of the better hot shoe flashes have the ability to do multiple flashes over a given time. Albeit at a greatly reduced power setting, which may not be enough for you.
The first flash I ever noticed that could do this was the (expensive) Canon 580EX II, so if you look up the user guide you can see the terminology they use. As well as the limitations. Once you can recognize the lingo you could look for the much-cheaper knock off units.
What I'm thinking is, for example, to set the flash to fire 50 times at 1 millisecond intervals, or something along those lines. So the flash appears near continuous for roughly 50 milliseconds. Now, the downside is that the power has to be set really low, maybe 1/64th or lower, and it needs a cool-down interval of maybe a couple minutes (?) between flashes, and I don't know if it's a smooth enough output to prevent "chatter marks" on the image. But... if it's anywhere near close enough, power wise, it might be feasible to use a bank of 4 flash units, giving a 2-stop boost over one unit. So maybee....??
Obviously (?) LED arrays are gonna be much more efficient battery-wise (electronic flash puts out a lot of IR, which is all wasted energy). But you'll also have to engineer a way to get even lighting, which i think is gonna be more difficult than meets the eye. Offhand I'd say that the most obvious way is to bounce it off a diffused surface (aka white paper), but this is gonna spread the light over a wide area meaning a big loss, maybe several f-stops, with respect to your subject exposure.
Anyway, best wishes with your attempts.