So, yes, I know that is one way to do it. What I do not know is how to inject
this into an existing dialogue between client and server. That is why I
thought a SIP proxy would be a suitable solution, because it would naturally
be placed between client and server to start with.
The SIP protocol does not allow you to inject messages in the middle of a SIP dialog. The CSEQs will be out of order on one end and other messages will be ignored. That’s why everyone points you to the b2bua solutions where you have different dialogs on both ends.
Start with accepting that, then look for a small b2bua that fits your needs, that isn’t configured to be a full featured PBX, which you clearly stated that you don’t want.