Thanks to both of you.

I think I am going to book my trip with Lufthansa or Air France because I can know in advance how much I will have to pay for my 2nd suitcase. This way, I won't become penniless at the airport.

I still don't understand why airlines don't communicate about their definite policy for excess baggage fees when flying multiple airlines on a same ticket. Or maybe there's no clearly definite policy ?
I had the same problem when I wanted to check-in a 2nd suitcase on a trip operated by Brussels Airlines and Jet Airways. I asked both and got a completely different answer...

I am going to study in Dublin so I can't just put the extra in my hand-baggage. It's not like I was going to a country where I could wear shorts and t-shirts only all year long.