There are 245 games in The Database where Black declines the Jerome Gambit with 4...Kf8, with White scoring 56%. To break that down, after the retreat 5.Bb3 (Houdini's choice), White scores 61%. The piece exchange, 5.Bxg8, has White score 57%. The complicating 5.Nxe5 scores 50% for White.
By comparison, there are 40 games in The Database where Black declines the Jerome Gambit with 4...Ke7, with White scoring a more encouraging 71%. Again, breaking that down, after the retreat 5.Bb3 (Houdini prefers the three retreats of 5.Bd5, 5.Bc4 and 5.Bb3), White scores 100%, as he does after the exchange 5.Bxg8. The complicating 5.Nxe5 scores 0%.
In summary, if Black does not take the Bishop in the Jerome Gambit, both computer analysis and game play suggest that White should either retreat it or exchange it, with good-to-very-good prospects.