Well I've found a resource that addresses phase noise in opamp systems:
www.ti.com/lit/an/sboa066a/sboa066a.pdf , and I think I'm starting to understand the sort of noise level I can expect from a given stage or system.
I've also found while wading through datasheets that going for high bandwidth/slew rate opamps actually doesn't help in my case - while they may be down to .9nV*Sqrt(Hz)... that's at 100kHz or more, and the figure balloons at lower frequencies. With that in mind, I set again looking for lower speed, low noise opamps, and ended up finding the audio-specific OPA1612 as being the winner - audio opamps are almost always specified down to 1kHz or 100Hz in their noise terms, and this one is typically 1.1nV*Sqrt(Hz) at 1KHz, 1.5 at 100Hz, and only 2 at 10Hz - the lowest stated noise I've come across yet in the audio band.
Perhaps more interestingly, investigating into Johnson-Nyquist noise from resistors..... the resistors used for gain in the opamp stages will contribute more noise to the system than any other part. A 1k resistor at just over room temperature will contriubte about 4nV*Sqrt(Hz), and a 100 ohm resistor is just over 1nV*Sqrt(Hz).
Since all these figures are for fixed voltages (and currents) relative to the frequency of the signal, I should also be able to lower the noise floor by using wider supply voltages (100 nV of noise is a lot more to a 1Vpp signal than a 10Vpp signal), but am limited by current sinking capacity and the fact that going with large voltage swings needs more slew rate to keep up and larger resistors (more noise again) to keep the current levels in check. It's also to my advantage to have very high gain on the first stage because the mic level input is so small (8mV is specified by the Rode NT1 datasheet) so that later stages' contributions to the noise of the system make less of an impact on the SnR.
Knowing this, I think my priorities for designing the circuit should be:
Fewer amplification stages, the first one with the majority of the gain
Low resistor values to minimize thermal noise (but not so low as to burn out my opamps)
Low noise audio op amps over low noise high speed op amps
A high speed, very low jitter comparator to square the fundamental frequency signal to the point that the frequency counter can trigger accurately on it
Put the whole thing in a freezer to further reduce thermal noise

As for the back-of-the-envelope calculations....
OPA1612 noise figures: 20Hz ~8nV, 100Hz ~15nV, 1kHz ~34nV, 5kHz ~80nV, 20kHz ~150nV
So on the input stage without figuring in resistors that's:
10^-6 range below 5kHz then 10^-5 range above that
And on a second stage brought to let's say 3Vpp
10^-9 range on the low end and 10^-8 range on the higher frequency end
Then a 250 ohm resistor to limit the 5V supply opamps to 20mA: 2nV*Sqrt(Hz) or
20Hz ~9nV, 100Hz ~20nV, 1kHz ~63nV, 5kHz ~141nV, 20kHz ~243nV
So considering there will be a couple resistors per opamp and one of them will be much larger, you can knock at least one digit off the noise floor estimates from before. Not exactly the precision I'd hoped for, but at least this is for single waveform period accuracy. There will be multiple waveforms to average and as the noise floor gets larger, you will get lots more averaging in a given gate time.
And that's just the signal path noise if I can keep the power rails completely still!
If this is the best that can be done without custom parts, thousands of dollars, or cryo cooling, I'm satisfied with doing it and seeing what sort of results I can get, but if you have any ideas for taking out or reducing some of the noise sources, I'd appreciate it. Getting a mic with an order of magnitude or more output voltage would clean up the first stage noise on the circuit, but I'm not sure if it's worth doing because the NT1 while having a very low output amplitude, is a very low noise floor mic - one of the best available short of a thousand bucks from best I can tell.