I have recently bought a fantastic February 1975 4001 Rickenbacker.
Would I risk damaging the instrument if I put on some Rotosound 45-105 flats?
HELL YES YOU WILL DAMAGE THE NECK!!! WITH THE INCREASED TENSION OF ROTO MONEL FLATS YOU WILL DESTROY THE NECK OF A 4001. I DID THAT ON ANOTHER BASS!!! I'm saving my money to replace the neck or truss rods on that bass, whichever my tech says is the most economical.
Pardon my shouting. Nobody bothers anymore to really get into the subject. It's not rounds, it's not flats. It is the inherent tension of any particular set. Now, that said, when I had a 1976 4001, I used GHS Boomers 45-65-85-105 with no issues whatsoever. I have a 1981 4002 (yes, "2," not "1" or "3") that I have used GHS Progressives 45-60-80-105 since I got it in the early 90's with no issue. If you want flats on a 4001, then the recommended set is the Optima RB, because it is the spiritual successor to the OEM Maxima flats that Rickenbacker used on these basses.
OK: all else being equal, a 45-100 rounds set will have about the same tension overall as the Optima RB 40-100 set because the flatwound ribbon wire has less gaps between the windings than the rounds, so they have similar mass, and therefore similar tension. By contrast, a flatwound string will have much more tension than a roundwound string of the same overall diameter because there is less gap between the windings on the flatwound, therefore more mass, therefore more tension.
That said, a well-maintained and properly adjusted 4001 with no internal damage to the neck from improperly maintaining the hairpin truss rods should be able to take about any reasonable set of round wounds, even in a 45's set, as mine did. But for flats, due to the increased mass for the given diameter, I would not put anything heavier than the Optima RB set or the Fender 9050L set, which is 45-60-80-100.