You could use the LFCLK as a clock component and rout that to a pin. Should work in deep sleep with 32kHz.
Then try to use a 31.25 kHz xtal and enable the WCO. Test if that runs correctly.
I understand the LFCLK is very inaccurate (60% tolerance), If I connect an external 31.25 kHz crystal to the micro then I might as well leave it on the external device as it is now and I still have the problem of the component becoming obsolete.
If not for the crystal becoming obsolete, I would not need to change the micro circuit. My thinking is that using the micro with a standard crystal (say 4Mhz) means I have a freely available crystal-based generation of the 31.25 kHz signal now but the current consumption will be too high.
Some of the BLE modules have a built in watch crystal on-board; You could try one of those and use the WCO, but the clock is at 32.768 kHz instead of the 31.25 kHz that you want. Otherwise, routing a clock to the output pin with a clock-divider component may work...