Am I correct in understanding that the pin "ICP1" isn't routed on the Arduino Mega 2560 board, so I can't capture the value of TIMER1 on an external signal edge? At least not without buying/building a different board?
You are correct, but note that ICP3 and ICP4 are available on the long "external I/O" connector as pins 49 and 48 respectively. These belong to timers 3 and 4 which are also 16-bit timers thus might be a good substitute for Timer 1 in your application.
-- The Aussie Shield: breakout all 28 pins to quick-connect terminals
Thank you, I missed that fact! That should work fine for me, as long as there aren't any weird restrictions using Timer3/4 as compared to Timer1. Off to the data sheet!