There are a ton of issues with the official CPI numbers that none of this accounts for and even the BLS admits to in a limited extent. It doesn't account for people buying less expensive substitutes when items get expensive, it only includes "staples" despite consumers potentially spend a ton on newer items, it doesn't account for quality changes of items, it has an undue weight on urban areas, doesn't account for shrinkflation or other hidden forms of inflation, and lots of others.
The BLS isn't independent. The commissioner is hand picked by the senate, and it's currently someone who's very deeply rooted in establishment politics and has close connections to the Koch family, the fossil fuel industry, and Trump.