There's a simple reason for that: The price of a Big Mac is probably driven mostly by volatile inputs like food and energy. Food an energy are only a part of the overall inflation picture, and they're generally not leading indicators of long term trends. There are two key CPI numbers: headline and core. Core removes volatile components like food and energy because they tend not to be built into long-term inflation expectations. When headline and core diverge significantly, the divergence generally goes away in the near future, and it generally goes away when the headline number moves back toward the core number, not the other way around.
Long term, inflation is driven mostly by expectations and the interaction between expectations and wages. Spikes or drops in the price of oil or beef tend to revert to historical norms, so while they make for interesting charts in the news, they're not really all that useful for long-run predictions because they don't really provide steady enough "feedback" to feed into slower moving prices like wages. The divergence between the Big Mac index and our other inflation metrics is likely driven by that phenomenon rather than an actual failure of the broader metrics.
Anyway, it's not a matter of "assuming" the BLS knows best. You'll find that people who actually study this stuff and use the data think they put out a very useful set of indices and have very good reasons to ignore outliers like the Big Mac index. The BLS basket of goods is very broad, well analyzed and completely public. Every quarter we hear the big headline about something like, "Chicken prices spiraling out of control! Inflation to come!" Not unless the public at large *really* eats tons and tons of chicken and the trend continues for some time.
There are other cross-checks that are pretty easy to do. For example, if the Big Mac Index was truly reflective of reality, we'd be seeing massive capital flight from the US to foreign markets with better inflation numbers. If you know the dollar is losing 10% in inflation every year, just sell your dollar-denominated assets, buy Japanese bonds with yen and enjoy your nearly risk free ~12% real return in dollars. Either the big money (who are presumably the puppet masters driving this whole scam) is too dumb to do this or that's really not how the numbers work out. Either that or all currencies everywhere are inflating at roughly the same rate without feeding back into wages anywhere, but that would be a really interesting macroeconomic state of affairs.