Why not arabic? French is kind of obsolete.
Mandarin would open the EU to the largest number of speakers. Spanish would be a good second, if English is politically untenable, but of course it's already an official EU language so choosing it might annoy the other states. Romansh is an European language not used in the EU; choosing it would level the playing field (and would not be so politically complex as another such like Basque).
No, making Mandarin the working language of the EU bureaucracy wouldn't open the EU up to any additional people, and the vast majority of Mandarin speakers are subject to one government or another that has no interest in the EU (and particularly, for instance, being accountable to the European Convention on Human Rights.)
> Romansh is an European language not used in the EU; choosing it would level the playing field
No, it wouldn’t. In terms of making communication within the levels of the bureaucracy to which it applies all but impossible, sure.
Not sure why that would be the goal, though.