From what I could find, it doesn't look like these types of boxes are listed for any type of fixture support.
When looking at a box that attaches to a stud/joist using nails (B520P), it states.
Suitable for fixture support up to 50 lbs.
With the the old work version(B618R), it states.
Not listed for fixture support in ceilings.
A box like this (BH614R), states.
Listed for wall fixture support up to 10 lbs.
The last page of this catalog shows some round boxes, and the weight they are rated to hold.
Based on this information, I would say the box you want to use will support less than 10 lbs.
The best options in your situation, would be to move the fixture location to allow you to fasten the box to a structural member, or add a structural member to mount the box to.
Update:
I contacted a company that manufactures these, and here is their response.
[Link to product omitted]
As shown on the link above this item is not listed for fixture support. We do have several other boxes that are rated for fixture support.
[Link to other product omitted]
The link for boxes rated for fixture support, pointed to boxes that attached to structural members.
A horizontal board between two studs is typically fire blocking. The idea of fire blocking is to prevent a cavity in a wall more than 8' high where fire can quickly spread. You'll see this in any room with over 8' ceilings. For it to work properly, you can't have any gaps allowing air to go from the lower cavity to the upper one. This means any gaps need to be caulked with fire rated caulk (that's typically the red or orange caulk). Why the heck someone installed this blocking and electrical box in the same place is bizarre, either one could have been moved up or down a few inches.
To work around this, you have a few options. You can cut a gap into it, and fit your old work box, but from the small opening you'll have a very difficult time cutting out the back. The result will require a load of caulk on the 2x4 that you squeeze the old work box into. And you'll need to install it a few inches to the side so there is some material continuous from side to side.
You could also fix this the right way and open the wall up, and reinstall new blocking a few inches up or down. Then you can remove the old blocking and install a new work box.
And the easiest solution is as DA01 says, install the new switch a few inches higher so it's above the blocking. I'd probably go with this solution myself, unless you feel like doing some drywall repair.
Best Answer
It also says (specs on Home Depot web site) not listed for fixture support in ceilings, so it isn't even a question of 5.6 < 6, but rather "not designed for this purpose at all".
I believe the problem is not "plastic" but rather the wimpy little "swing clamps" - by design then can't support much.
The solution is a box that is actually nailed or screwed into supports. There are plenty of options like: For a fan, you really need to make sure it is securely attached to a support (e.g., stud). For a light, there is a bit more leeway as it is (usually) not as heavy and does not have the constant motion/vibration of a fan.