Shadowstats is complete garbage. Their "alternative measurement" of prices is to take the BLS numbers and add 3.5% to it. That is it. See "The [$175] subscription price of hyperinflation-predicting ShadowStats hasn't changed in 8 years"[0] from 2015. In 2024 it is still $175, so make that 17 years.
If you want an actual alternative to BLS numbers, there is the https://thebillionpricesproject.com/ from MIT.
[0] https://theweek.com/speedreads/449820/subscription-price-hyp...