UPDATE: With over 60 days (statutory maximum) having elapsed since I submitted my online renewal, Permitium was still showing my application at the "“assigning to processor shortly..." stage.
Yesterday I emailed and called the DPS CCW unit to chase my renewal through the system. I had to sit on hold for 30 minutes as there were 15 other people ahead of me in the queue. When I got to a human, she was very efficient - I just gave her my SSN and name, and she said she would immediately send my application on for review. This mirrors my wife's experience last year - the process moves at a glacial pace unless you call and check status, then it miraculously gets approved.
This morning I got a text and email from Permitium confirming that my permit has been approved and will be mailed to me.
No reply yet from my original inquiry email, but they may not reply now as it is moot.
I will update here when my permit actually arrives.