Well I did some testing and some playing around, and after a few hours of fine operation I think I've got a good replacement.
After considering the options to replace the crystal, I looked into buying a programmable oscillator and then placed an order - Digikey would sell me a programmed SGR-8002JC-SHC 5V oscillator for like $4, so it was certainly cheap enough to attempt.
While I waited for the part, I test the crystal on a VNA.... and it looked fine, so I guess this issue is the degraded feedback in the mystery chip, which bodes ill for the longevity of the system in case something is slowly failing open.
I tested with a signal generator what level and connection I needed, and while it looked like 5V would sometimes work, it needed a 5.3V amplitude or so on the sig gen, and it needed to be applied to both connection pins for the crystal to work reliably. Playing with the phase on the second channel (the other pin) didn't seem to help anything, so I decided with the oscillator arriving I would build up a little mod board and just AC couple the output into both pins (since the pins are biased by the micro).
The oscillator arrived and I built up a little board that barely fit in the space where the relatively large crystal was. Since there isn't actually a 5V rail in this camera, I included a TO-92 LM78L05, some bypassing, the oscillator, the AC coupling caps, and a filter.
The filter was made just through trial and error, and since I didn't know the impedance it was driving, to be sure the amplitude was in the right ballpark, I built in a load resistor and took a look on the scope when I was powering it up and making changes out of circuit. I ended up with a series 22uH inductor (15uH was more efficient for power transfer, but then had a peak to peak voltage above the supply rail of the chip it was driving), about 2000pF of shunt capacitance, and a 7.5k Ohm load resistor. Then I just used a couple of 0.1uF caps to AC couple into each side. I played briefly with some copper tape for a shield, but grounding to the board itself stopped it from running, and grounding it to the board ground plane let it run for a bit, but then shut off. Since the camera isn't really grounded (isolated wall wart) and I wasn't really seeing much noise on the output, I decided to just leave it off and call it done.
I can definitely recommend buying a programmable oscillator to build into a board to replace an oddball crystal value, they were surprisingly cheap, could be programmed precisely (they gave me 1.29155MHz on the nose!), and are small enough that with a more compact PCB could replace relatively small crystals (maybe not SMD or short can oscillators, but certainly the taller standard can size or larger), you just have to be able to build up your filter to make sure the drive voltage and waveform is good.