As the subject says it is possible - ignore the naysayers
It is just none obvious and certainly does not need a reinstall
The single box appliance works by hiding the idp behind the gateway.
To do this it puts it into a reverse proxy called NAM-RP with a proxy
name of NAM-Service
So all you have to do is change NAM-RP to suit your needs.
Change the published DNS name and it will change the Base URL
Change the certificate to a suitable one under Web Servers
Last peace of the puzzle came from support (thank you Brent!)
The cookie domain defaults to - it needs changing
Don't forget to also change to port to 443 instead of the default 8443

I did this to change from to
together with a wildcard certificate to *

Works like a champ

This would explain why there is no documentation to explain why you
can't change the base url - because it is simple to change once you know
the IDP is behind a proxy.

Hope this helps someone.

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