I think that definitely it is possible to do a "transfer", although that probably would imply some extra overhead to deal with the requirement fulfilling of your destination university.
I started my PhD in a European university, to move to a US one. I said this because I was in a similar situation and I think that talking openly to your advisor is a key thing. You said:
My relationship with my advisor is excellent.
I think that if this is really the case, you should openly talk to him for the following reasons:
- If your Prof. is reasonable (I am sure he is given your words) he will understand your willingness and ambition and will be able to recommend you what is the best thing to do. Present to him the reasons as honestly as you are doing in here, he for sure is aware of situation.
- He will hopefully help you through the process with strong recommendation letters and with recommendations to european colleges that he knows, increasing heavily your chances of landing in a good program.
- Offer to keep him involved. If you consider this is a good opportunity for you, it may involve being a good opportunity for him too. Try to keep him in the loop -- maybe he can be your PhD co-supervisor?
- Talking to him, if you do it carefully, does not mean burning your boats of staying in your current institution. Once again, assuming he is a reasonable and wise person, if you tell him that this thing is going in your head, that this idea came to your mind and ask for his opinion he will most likely do not consider it disrespectful (in fact you are being honest with him) and he will in my opinion do not prevent you from staying if at the end things do not work out.
Best of lucks!