Thanks for the quick reply! That's the part I was missing, I forgot the proxy was named in the app's config.json and not just the proxy file.
Update in case it helps anyone else: While I was no longer prompted to login, my print service still wasn't working, until I remembered I needed to configure it to print the secured services in my web map.
Print maps that contain secured services—Documentation (10.5) | ArcGIS Enterprise Then I saw that my vector basemaps weren't printing, until I redirected the app to a newer JS API that supported printing them.
Oh, and in regards to your original question: yes, I do sometimes see a login screen for the proxy still. Refreshing the page fixes it, however that's annoying and looks really bad for our customers who probably just navigate away from the page assuming it's broken. There's a thread here, but it doesn't seem to have been completely fixed yet Login prompt appearing randomly · Issue #300 · Esri/resource-proxy · GitHub