If you use an outbound proxy for the endpoint, you also need one for the address of record. This and your contact setting may be responsible for the incomplete looking request URI (INVITE request_uri).
It’s usually difficult to see what the correct settings are, since one does not know how all the fields are evaluated by the service provider. These guys usually have some recommended configurations, which one should read.
I am not sure whether the loose routing parameter is actually required these days, as it now seems to be the standard instead of working with hop lists. At least I have a couple of service providers, where it doesn’t matter whether you specify lr, lr+hide, or nothing.