My mistake, the IE debugger wasn't showing the POST... When I looked through Fiddler, I could see it as a POST.
For some reason the query is failing at the SDE level and my failure to see the POST in the IE debugger was leading down the wrong path.
The permissions are set up correctly, the layer draws correctly, the service sublayer is available for query on the REST page, but when the query is submitted, it takes far too long (much longer than it should). The same query works fine through arcmap, and the code I have to set up the query works fine with other layers and services.
On to the Rest or SDE forum I guess 😉