I am looking for a European bank offering a Bridge Loan for a
commercial real-estate project. The amount we need to borrow is euro
10 million.

Greetings,
As a onetime banker in Germany, I would suggest that your best  - only
-  bet is to look to banks in the country where the project is.  A
bank in another country will almost certainly refer you to their
associated bank in the country or to their own operation there.
If you have the longterm financing already and if it is also from a
local bank, unless it has already showed disinterest in helping you
with the bridge financing  - either itself or with references -  I
would talk to them.  The people doing the bridge financing will be
wanting to know all about the final loan that will pay them off, so
they will be wanting to contact that lender anyway.

I hope this suggestion is of some help.
Good luck, Myoarin