Yes, I see the problem now too. Really dumb design they made to not have allowed it to work with either box. The fact that it puts part of the fixture into the box raises another interesting question. Despite what they claim, can this be installed in a regular ceiling box and be code compliant? There are box fill rules and when you start using the box for the fixture itself, it's going to use up some of that volume, especially when per the new drawings, the fixture consumes the entire surface area of the box dimensions. If it extends much at all into the box, I would think an extra deep box might be required to be code compliant. So, since he needs to install a new box, I'd get a deep one, if possible.
Yes, and just so he understands, if he does cut the cable, then he has two choices. Put another ceiling box nearby that the undamaged cable can reach to, then run a new cable from there to where the light is. That leaves you with a new junction box you have to cover with a blank plate, because it has to be accessible and it doesn't look pretty. Whether that is even possible, depends on how the cable that you can't see runs. If for example it comes from above, straight down into the old box, then you don't have means to do the above. If it runs across sideways, then you probably can, assuming you can free up enough cable. And that could easily involve opening the drywall.
Option two is to trace the cable back to somewhere that a new junction box could be added and remain accessible or where it's connected in an existing box somewhere. Adding a new box like that could be fairly easy if there is accessible attic above. If there is a second story above, etc, then it's usually a nightmare and you're back to option 1.
So, before I did try to cut out the old box, I'd be 99% sure I'm not going to damage the cable. Which, may be a good reason to consider just opening up the drywall to do the work that way. As I said before, it's going over a sink, and many times that area is it's only it's own little section of drywall, so you don't have to then paint the whole kitchen ceiling, etc. The tradeoff is that with that approach, there is guaranteed more work involved than if he uses the cut it/tear it out method and it comes out fairly easy.
Personally, the first thing I'd do is do a search to find out if there is another similar LED light available that doesn't have this mounting problem. It's not an inherent problem that would be common to all lights, just this one did a bad design. Or perhaps consider other lighting options. It's a neat light, but it does have one big drawback....
Turn the power off to that fixture before you do it, for obvious
It's an LED light so it only weighs a couple pounds, if that. As I said before, there are old work boxes similar to the type you're talking about that are rated for holding fixtures up to some number of pounds, eg 3, 5 etc. Instead of the swing out ears they have metal side pieces that expand outwards. I wouldn't use them for hanging anything of much substance, but they are perfect for that small LED light.