A lof of actual money rides on CPI measures. For example social security payments are indexed. Which is one of plenty of strong incentives to keep the "measurements" low.
BLS is part of a large organization. Specifically it's part of the United States Department of Labor (DOL), one of departments of the executive branch. I haven't yet heard of a government with an incentive to find a high inflation number.
(And if it were an "independent" agency like statistics are in some countries, that would provide more shielding but not necessarily perfect either.)
Asking for specific evidence of a specific BLS personel filing a grievance or similar because of pressure is placing the bar way into the "blind" territory. This is not how influence works. I mean sometimes it is and we get to see it laundered in public, but that's not the common way. In the case of BLS it may be there, I haven't searched.
