Rust is actually right, you cannot have multiple mutable references to a Database object without things going down the drain. (This is related to the fact that, like other comments said, &mut is an exclusive reference).
However, achieving something like what you want is still more than possible in Rust. You can do this with the pattern of 'interior mutability', which in its simplest form is just a Mutex. This allows upgrading a shared reference to an exclusive reference, so that you can safely mutate an object while upholding the expectations that a mutable reference is exclusive, and a non-mutable reference does not change from under your feet.
Of course, for a database, you will probably want a more advanced implementation of interior mutability, so that you can commit multiple transactions at the same time. (Or not, it seems to work quite well for SQLite.)
However, achieving something like what you want is still more than possible in Rust. You can do this with the pattern of 'interior mutability', which in its simplest form is just a Mutex. This allows upgrading a shared reference to an exclusive reference, so that you can safely mutate an object while upholding the expectations that a mutable reference is exclusive, and a non-mutable reference does not change from under your feet.
Of course, for a database, you will probably want a more advanced implementation of interior mutability, so that you can commit multiple transactions at the same time. (Or not, it seems to work quite well for SQLite.)