Well, I do think there shouldn't be anything preventing someone from using the Aile, Launcher and Sword Strikers at the same time. It's probably not the smartest thing to do, because despite the auxiliary batteries you will drain the mobile suit's power reserves extremely quickly. Which is why the AQM/E-YM1 Multiple Assault Striker includes four extended battery packs. You also weigh down the mech with all that extra equipment.
You can see a Gunpla using all three Striker packs here:
https://www.bing.com/images/search?...c=U316&ajaxhist=0&vt=4&eim=1,2,3,4,6,8&sim=11
There is also the Multiple Assault Striker and ISWP Striker, which combine the functions of all three original packs.