Interesting retailer you have there... it's the same setup as the THIS ... Birdy offered the same setup for the legacy frame bikes (which I prefer) originally as the Sachs 3x7 and later the SRAM Dual Drive .. I'd look at dealers who are Birdy savvy ... I have a brand new complete Birdy wheel with Sachs 3x7 , cassette, shifters etc. in my parts department for under $300 complete ... shipping to France would probably be prohibitive, but that gives you an idea of how inexpensive the upgrade would be... old wheel out, new wheel in, add shifters and cables and adjust... pretty easy... I used to ride a Birdy Silver, same bike as yours, but with the Sachs 3x7 .. one of my favorite bikes...

sash
 

Come to think of it, I've got a spare sash 3X7 for £25 on ebay. Its in bits at the moment. But it was working fine before.
I prefer the sram version as it shifts under full load.
I Think if I still have my Birdie next summer I will put it on.

Thanks much for the infos. I'll stop at the store next time I'm in the neighborhood and ask them about installing either the Sachs 3x7 or a SRAM Dual, and see what they say.