I had to step away from the project for a couple of days to handle other things. As additional information, the PIC32 I am using is a PIC32MX110F016.
This morning I modified the code so I am using the Output Comapare function of the PIC32. According to the data sheet, when the count in the Output Compare register matches the value in the Timer2/3 register pair, the OC output pin gets set with no cycle delay. I am using the Timer4/5 pair for the frequency counter input.
I put my programmer in debug mode and confirmed that is true. When the OC match occurs, I use the OC output pin to disconnect the 10 MHz signal from the counter input pin using a TS5A3159 analog switch. When I stop the program inside the OC interrupt routine, the TMR4 register contains 40,000,000 which is exactly one second. If I set Timer2/3 and OCR1 to 400,000,000, that is what I see in TMR4 register inside the OC interrupt. So there is in fact no cycle delay when the OCR1 count matches the PR4 value.
The counter has been running in the 10 second mode for a couple of hours and constantly displays 100,000,000. About once a minute I see a count of 99,999,999 due the input signal not being edge aligned with the start of the delay period. But that is no big deal.
All I need to do now is to find someone with a $10,000 HP or Tektronix 12 digit frequency counter who can tell me the exact frequency of my GPSDO. If I knew that, I could put my counter in a 1000 second count mode and see how it compares to the expensive counter.