I wonder if you could get something like in the United States.
The Federal Circuit Court system was established in 1789, which predates Washington DC. Maybe instead of establishing DC, it could be decided that the Federal government could also travel between locations, to ensure equality between all parts of the country. In theory, they could even travel around the district like the circuit judges themselves did, but that seems a stretch.
The whole apparatus could relocate every 2 years or so; in some sense, that would only mean one capital at a time, but in another, it would mean 6+.
Note that when first established, each Circuit had 2 Supreme Court justices permanently assigned to it, and they were expected to spend a decent amount of time hearing cases in these circuits, so there is some precedent for the idea of directly involving the Federal government in this system.