The government already publishes separate indices for a few hundred separate spending categories, ranging from pet food to parking tickets[1]. The reason nobody bothers to do what you suggest is that it's a ton of work for pretty much zero return: the resulting numbers are not actionable and won't change anybody's opinions from what they already believed to be true.
Granted, the BLS indices are not raw prices. But using actual raw prices is a few orders of magnitude more work, and the results are rather unspectacular (if you summarize the price changes into lower dimensions you get similar results as what the good folks at the BLS already did for you) [2]