I've never noticed this problem with my FR2LE, but then I always use high output mics & never need to switch in the +12dB gain. But, in the interests of science, as they say, I've just given it a go. With 150R resistors between XLR pins 2-3, & the gain & level cranked up to max, I neither hear any periodic noise nor see any sign of it on the spectrum. I simply get white noise at -65dBFS (Left), -66dB (Right).
When recording, the "REC STBY" LED flashes in unison with the CF card write cycle, so there's actually a lot happening inside the machine at this point.
How old is your FR2LE, btw? My machine is quite old, about 11 or 12 years. s/no 0960847TE, but I've had no probs. If the noise had increased over time, I'd suggest that possibly the power supply filter capacitors had deteriorated. Other than that, it's hard to say.
In my test above, the FR2LE (v.1.30) was running stand-alone with a 7.2V RC battery pack. If you have power supply &/or USB cables connected, it might exacerbate the problem. Bit annoying really, as the front end of this machine was really impressive for its time (& still is). It's the user interface, h/phone & main outs which let it down.