Late to the table, but FWIW, many calipers are designed with more of a flat/U at the top where it meets the tyre so may well have clearance, like these SRAM Apex. Not sure where you are based, but in Europe they are only 20euros.
That may well work. Looking at those calipers too, the outer arm appears much lower than the pivot/mount. That would suggest the brakes are quite far from the rim?
I do think another caliper may work. I think the brake design is a little flawed in how it is designed. I can see how you would have plenty of clearance with the frame.
