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@Tea_Bee
It's odd that you need "the shortest cranks ever made, ever" - after all, you've only dropped from the original 630mm rims to 622mm (700C). So your bike is just 4mm closer to the floor than the original design.
You can get (junior) cranks down in the 150mm range but unless your current cranks are super-long, it sounds more like your cornering technique on fixed as much as anything. Personally, I love canting the bike right over in corners and keeping my body slightly more upright, but I've had to do the opposite when riding fixed and keep the bike more upright to maintain pedal clearance but lean my bodyweight across instead.
Instead of shorter cranks (or even as well as) you could try running with wide slicks, like lots of people are doing. That would jack your ride height up by double what you lost in the change of wheel. Good rolling resistance on the wider tyres and a softer ride. Can look a bit strange on old racers though.