Just a quick update, and maybe a helpful suggestion for others.
I ended up hard-wiring button pairs in parallel and attaching them to the same port :
Controller 1 (numbering according to the above diagram)
J1_8 and J1_5 => left shoulder
J1_9 and J1_4 => right shoulder
J1_7 and J_11 => select
J1_6 and J1_10 => start
J2_6 and J2_8 => select
Thanks to everybody for help!