Hi Jeff and Timo - thanks for replying! Your comments definitely helped us out in our troubleshooting. Now, I'm not the technical person (anymore) so apologies if I don't explain this well.
We're actually working with two separate clients with very similar deployment patterns - and we noticed this actually happened to both.. (we're still in build mode for both - not in prod yet). What had we done or changed that had actually been done for both clients?
The only thing we'd done consistently was to implement the recommendations of the ArcGIS Enterprise Hardening Guide.
The team has identified that one of the changes - specifically enabling this setting:
So whatever this is doing (as I said not technical person!) is causing us to be prompted again. We have unchecked the box on test environment to test, and the usual expected behaviour of not being prompted again is back. So, this will end up being a conversation with our client about their preference - security or user experience!