Here are English translations of relevant papers of Laplace (search for "Bouvard"). Laplace appeals to the next century of data. It's not phrased as a bet there and while I can't address everything he wrote, older translations use "bet" where newer ones use "odds."

Newton used (his derivation of) Kepler's laws relate the mass to the period of satellites; I think Bouvard did the same. Newton correctly calculated the mass of Jupiter because it has easily visible satellites, but was off by 15% for Saturn. Bouvard differed by 0.5% from Newton on Jupiter, but both were 2% off (Laplace was overconfident). Bouvard's error on Saturn was less than 0.5%. Since Laplace thought he should be less accurate on Saturn, it was probably luck. Bouvard also computed the mass of Uranus within 30% by looking at its effects on other planets. ETA: since Laplace says the mass includes the moons, it probably isn't computed from the orbits of the moons; perhaps it is the effect on other planets.

Calculated from gravitational force.