Three teams cannot make a bracket — someone gets a free pass to a two-team "tournament." Round robin is the only honest format at three: everyone plays everyone, three matches, done. It has one famous flaw, covered below.
3 teams · 3 matches · one idle per round
Three matches: 1v2, 1v3, 2v3. Only one match can run at a time no matter how many surfaces you have, and one team is idle in each — give the idle team scorekeeping duty and the format runs itself. Three games at 30 minutes is a tidy 90-minute event.
The three-team round robin's known failure mode: A beats B, B beats C, C beats A. Everyone is 1-1, head-to-head is a perfect circle, and no rematch schedule fixes it in principle. Decide the tiebreaker before game one:
Genuinely: the jump from 3 to 4 teams doubles the games (3 → 6), removes the idle team, and makes the tie math far friendlier. If a fourth team exists, get them in and use the 4-team schedule. If not, three works — just with scores recorded.