The plots aren't right because I took the two-mirror mode spacing formula from Kogelnik and Li without applying the necessary modifications for a 3-ring cavity. The correct formula for mode (m,n) is $f_{mn} / f_\text{FSR} = (q+1) + (m + n + 1) \arccos(1-2L/R) / 2 \pi + \eta /2$, where $q$ is the axial mode number, $L$ is the half of the round-trip length, and $\eta$ is 1 if $m$ is odd and 0 if $m$ is even. (Note: for a 4-mirror cavity, $\eta$ is 0 always.) For reference, the K&L formula for a two-mirror cavity is $f_{mn} / f_\text{FSR} = (q+1) + (m + n + 1) \arccos(1 - L/R) / \pi$, where $L$ is half of the round trip length.

Instead of making more scatter plots, for each value of $g$ I computed the distance (in MHz) from the fundamental resonance to the nearest HOM resonance (up to order 20); the result is shown in the first attachment. I then picked the most promising $g$ factors and simulated a frequency sweep across a full FSR for a 3-mirror cavity with $L$ = 20 cm and $F \simeq 300$; the results are in the second and third attachments. Each mode is labeled with its order number, as well as 'e' or 'o' depending on whether $m$ is even or odd. I picked a arbitrary uniform amplitude for the HOMs, so these plots are only meant to indicate the locations and widths of the resonances. I've spot checked these plots against a Finesse model, so I'm reasonably confident that I've got the formula right this time.

I think the moral here is that the nearest HOM resonance is going to be about 16 MHz away from the fundamental, assuming $L$ = 20 cm. If we make $L$ = 10 cm, we can get to 32 MHz, but (depending on how bad the intensity noise at 30 to 50 MHz is) this potentially requires increasing the finesse to something like 600 to get the required intensity filtering.

If we go with a 4-mirror cavity, the modes don't have this $\eta$ degeneracy breaking, and there are $g$ factors for which the nearest HOM resonance is more like 30 MHz for $L$ = 20 cm. I have plots for this, but I want to check them against a Finesse model.