Suddenly I am very interested in your project, the church I go to has quite a bad time with feedback and this might offer some gain boost. 
Still think the digital approach is the better for the LF oscillator as Kleinstein suggested using a DDS chip or module, you can get versions with quadrature out, but then the linear option is direct and simple, and small mismatches shouldn't be noticeable as long as you don't get a significant beat frequency.
According to the reference project, RV1,2 are meant to minimize any residual amplitude modulation at the output from the oscillator. If the quadrature isn't quite 90deg, or the multipliers aren't summed exactly equal, then the overall gain can be modulated slightly at twice the oscillator frequency. Those two pots are essentially to account for analog not being
exactly equal in their respective parts of the circuit. Neither would be needed for digital because that
is exact. (well, exactly equal to
itself anyway, which is what matters here)
Cyril Bateman: https://linearaudio.nl/sites/linearaudio.net/files/Bateman%20EW%2001%202003%20mar2003%2010%20to%20100uF%20caps%20and%20100%20Hz%20measurements_0.pdf
Has some nice articles about distortion of some Aluminium caps, they are much better than I expected, but one result was that Bipolar Electrolytics are always lower distortion than Polar Electrolytics of the same value.
Well, seeing as all of them had their highest harmonic below -90dB, I don't think I'm going to hear any of them.

Just stay away from tantalum, which I was anyway. (too many horror stories...)
If using a processor, you would need a 32bit core, probably ARM. I feel like an NXP evangelist(I really have no relation with them), but the iMXRT1021 would be a good starter. The development board(not expensive) even comes with an onboard stereo codec...
I'm more torn than before now, between keeping it analog and solving those problems, or making a DSP to do it. I do have some more ambitious projects on the shelf that will definitely require an audio DSP. In order that I was planning to do them:
- Stereo USB sound card with a signal path of: highpass -> gain -> limiter -> crossover -> individual gain -> corrective EQ -> anti-clipping limiters -> TDM -> class-D chip amps to a custom 2.1 set of speakers. Everything before the crossover is user-adjustable; everything after and including the crossover is tuned for the installation and left alone. (and of course some blinkenlights to show what's going on, like HPF bypassed, limiter active, etc.)
- 8-in/8-out USB sound card plus 6 analog channels each direction (3 stereo in, 3 stereo out), with mic and speaker correction, mixing, and hearing/clipping protection. This one is going to be especially interesting in that the mics and speakers are going to be physically close to each other with foam, resin, or some other kind of barrier in between, and their primary purpose is to fill in what the unavoidable acoustic lowpass cuts out. So there may or may not be some feedback prevention in there as well, and I really want to keep the analog-to-analog latency to an absolute minimum to avoid a comb filter, which probably means writing my own DSP code for one sample at a time instead of the 64 or so that most pre-fab DSP's do that I've seen. (Analog Devices has a wonderful DSP chip and matching free IDE, if it weren't for that one specific problem for this particular application that I want to build up to.) The USB connection in this case is just to monitor what's going on at various points for debugging and possibly recording on a PC, and to read/write coefficients and possibly reprogram it altogether.
It would definitely fit to insert this SSB shifter as #0 in that list of DSP projects, but I can still see some reasons to keep it analog too...
...but I have no idea how processor intensive the Allpass filter network would be to implement.
Keep us up to date on your progress and results.
Well, there's these that I already have bookmarked:
https://en.wikipedia.org/wiki/Digital_biquad_filterhttps://www.musicdsp.org/en/latest/Filters/197-rbj-audio-eq-cookbook.htmlhttps://www.musicdsp.org/en/latest/Filters/64-biquad-c-code.htmlhttps://www.musicdsp.org/en/latest/Analysis/186-frequency-response-from-biquad-coefficients.htmlhttps://www.musicdsp.org/en/latest/Analysis/108-magnitude-and-phase-plot-of-arbitrary-iir-function-up-to-5th-order.html#commentshttps://www.musicdsp.org/en/latest/Effects/204-simple-compressor-class-c.htmlhttps://webaudio.github.io/Audio-EQ-Cookbook/audio-eq-cookbook.htmlBasically, a lot about biquad filters, how to write one, and how to calculate their coefficients from the desired behavior and back again, plus one example of a compressor. (A compressor with infinite Ratio, instant Attack, and finite Release, becomes a limiter, which is what I'll need for the other two projects.) The equations for biquad coefficients are not intuitive, just like every other DSP filter, but they're conveniently given on those pages for each of the filter types that might be useful. One of those given types is an allpass.
So, each analog APF (U2..U5) that isn't involved with the quad oscillator, represents one biquad with presumably constant execution time regardless of its coefficients (I'll need 8 of those, exactly as if it were a stereo 4-band full-parametric EQ, fed in mono; only the coefficients are different, and they're not equal between channels like they would be for a stereo EQ), the multipliers and the summer become trivial (one instruction each), and the oscillator is pretty straightforward too. If I'm using a 32-bit chip just for this, then I'll probably have enough overhead left over to actually do the trig functions instead of a lookup table.

All that's left then, is to connect the ADC and DAC to a world where full-scale is +/-15V peak. Pretty much every datasheet for those chips has an example circuit to do that, with some being more complex than others.