One way to run a plutocracy that was openly so would be with a ruling council that has trappings of a republic, but wherein you literally just buy your seats. There are some number of seats. Pick any number that seems useful to you, or is a cool "significant" number (7, 13, 108). More seats means a larger overt aristocracy; fewer means definite rulers rather than a ruling class.

Terms on the seats last for some number of years. At least 1, probably no more than 10. Maybe they're even staggared.

When a seat comes up for re-election, it's not really an election: it's a bid. Everyone who wants the seat bids money, and whoever bids the most wins the seat. The money goes into the national coffers. The council or senate or whatever it's called then gets to be the body that proposes and passes legislation. Maybe there's a prime minister who is whoever paid the most for his seat.

There could even be multiple levels. A lower council with more seats, and an upper council with fewer, and wherein laws must pass both to go anywhere. So the lower council seats tend to be cheaper, just because there are more of them and thus there aren't as many bidding on them.