So I have just tried it...the power saves file is different from the save data provided by homebrew. I was able to successfully install hacked files by homebrew, and then back up that file using powersaves, but I couldn't transfer it to another Smash cart.
Basically, the file backed up by powersaves is for your cart and your cart ONLY. And as you can't make custom codes through power saves (because that would be a useful feature), there's nothing we can do unless we can convince Datel to add the SLAJ equipment as a specific set of codes.
I will continue doing google fu to see if there's another way, but for now it looks like the powersaves route is a dead end.
EDIT: I don't think it'll go anywhere, but if they see there's a lot of demand for this we might be able to get them to make the codes.
https://powersaves3ds.maximummemory.com/coderequest/
It does require you to have the device (supposedly), but I asked them for a code that generates this equipment with compatibility for online play:
Smooth Lander Brawn Badge, 6 Attack, -22 Defense, 0 Speed
Anchor Jump Agility Badge, -6 Attack, 0 Defense, 50 Speed
Super Final Smasher Protection Badge, 0 Attack, 22 Defense, -27 Speed