I ended up speaking too soon about the success of the external monitor. My LTM runs the script from the SSH console w/out a problem at all. When it's applied against the pool via the LTM GUI, the node in the pool is immediately marked down. I opened a case w/ F5 support and also grabbed two packet captures: one from the LTM when I run the script via the SSH console (I see all the HTTP traffic generated by the script), and a second from the LTM when the monitor is applied to the pool (which shows no HTTP traffic at all). Perhaps the script isn't running when applied as an external monitor, though I am 100% certain that I've typed the script name correctly.
It's extremely perplexing! Hopefully an answer pops up soon that I can share with everyone.
Nimbostratus