I don't assume the government will be more reliable, because there usually are few direct consequences if the government source is wrong. So while I generally assume they'll try to do a good job if not pressured too heavily for political reasons, I don't assume that they actually have. In contrast, a commercial source whose data is too wrong to be useful tends to suffer. It's a much higher standard to meet--but, of course, the standard is to make money which isn't always aligned with accurate data.

There are piles of exceptions, though, so I tend not to assume anything about who is better and instead directly judge methodology (which, it's true, is more likely to be honestly reported from a government or academic source). In this case, BLS has decent but not stellar methodology for answering questions like these, so I think it's more likely that they are wrong, but more than anything I think that I don't actually know and also don't have any good ideas about how to find out. Which is frustrating, because even though the fractions are small either way, 4% and 10% are pretty different from each other.