I’m sorry I was not clear.
From my experience, calling polycom-chk-config from within asterisk doesn’t actually do a full reboot of the phone, only has asterisk tell the phone to update it’s configuration from the server. I believe this is what the system does when updating a template and choosing the ‘Rebuild & Apply’ option.
Apparently there is some kind of intermittent problem when doing this. Using the asterisk command to update the phones BLF field descriptions when people changed would randomly cause display problems after updating templates. The last time I updated a template, “Ellala” was displayed instead of “Ella” on the phones using a template I had changed.
There was no way to know that this happened without walking around and looking at phones or getting a call from a user. The templates was right. Most templates I changed to be ‘Ella’ reflected properly on the phone’s BLF button label, but phones using one template showed ‘Ellala’. The previous person’s name had no ‘la’ in it, so it wasn’t a holdover from them. Having asterisk tell the phones to update their config did nothing. Updating & applying the template again was ineffective. Rebooting the phones from either the web GUI or the phone itself fixed it. I’ve no clue what caused it or was able to identify any pattern to either how it would corrupt the button description or when it would do it.
This is why I continued hunting for another way to reboot them. (I really have no particular preference in using one tool over another, just needed one that works.) You mentioned having “the appropriate notification entry in your sip config files” - maybe I don’t have the proper config?? 
Cannot say without more details, but focused in on using curl (as I’d found several references to using it online that worked with older firmware, plus it’s been a good tool for me to automate mindless tasks to feed data to web servers in the past) and powered my way through the goofy requirements Polycom introduced sometime before the 5.9.8.5760 firmware & now have a functional script. 
(I actually figured it out several days before your original post and just now have had time to circle back to update this thread with all the details of the solution I found.)