Bonjour, Aur0ra, and greetings to you and your first posting on FT as well as the British Airways den of iniquity! Your question is a good one, you are most welcome here, so please stick around - French students is a demographic we want to encourage.... In addition to the friendly ami(e)s online, the team of ambassador is at your disposition.
The email from BA is the formal and worst case scenario. My suspicion, which I cannot guarantee, is that when you get to St. Exupéry airport you may end up just paying the BA segment and getting away with it, but given that the staff there are outsourced here are some tips: shuffle the heavy items around, can you get a few kilos off the lighter bag by taking items in hand luggage (BA is not quite a rigorous as Ryanair in that respect)? Stuff the coat pockets with electronic gear. Find the oldest check in staff member, be extra friendly and sweet, ask him/her to check it all the way through. Though this won't help with the Aer Lingus section, it is worth remembering that Club Europe tickets are allowed 2 bags. It is unlikely to be economic, but it is worth asking if there are any cheap paid upgrades available before checking in. Also Club Europe tickets that involve a Sunday night can sometimes be cheaper than 2 single Euro Traveller tickets.
In the medium terms this won't happen so much since BA is taking over BMI's Dublin service in the autumn, I guess this flight is well before then.