Hi,
a few questions first: 1) It's not clear to me from your description, does the fail occur from cold - initial turn on or after it warms up and you manually do a self test from the menu, or both?; and 2) When it fails what happens - does it not count frequency or does it not count interval or both?
If it doesn't count or do interval then I'd check the +/- 7.9V rails in the trigger circuit. If it counts but doesn't do interval then I'd be looking around the ADC/DAC in the interpolation section, I don't believe they're used in the count mode. As you indicate it is intermittent then perhaps it may be a dry joint on one of those IC or maybe tin whiskers around those IC (assuming it's lead free solder). My tin whisker remedy, and general FOD mitigation, is either an esd brush, or a soft toothbrush, while using a esd wrist strap.
Perhaps others more familiar with the 53132a workings may offer some different advice to get you started?