Or book with your AA number and carry your CX card with you. That way you'll get the CX benefits but won't have to remember to change your number before you fly. I do this all the time with QF and AA and haven't ever had someone question it, even when using CX or BA lounges.

As for whether to shift, remember that gaining EXP on AA will net you 8 eVIPs per year - quite valuable upgrade instruments for your European flying. And while you're waiting to get EXP the hard way (the only way to do so), check out the AA Platinum challenge that will help you get status quickly.

Peace.