But that combined with the empirical fact that rich people do buy things that make it obvious how rich they are

Wrong. People who want other people to think they're rich engage in conspicuous consumption. Actual rich people (at least first-generation rich), not so much.

For example, you could give people a $1,000,000 income but stipulate that they had to give it all away to charities of their choice, and make it near-impossible for them to reliably tell anyone that they had done this.

That would be quite useless, if you haven't first determined whether it's relative happiness increasing relative income, or vice versa.