LEDs and high currents
I would try something like this. You want the new circuit to have roughly the same characteristics as the chime. I don't know enough about the details to give exact values. 

The value of R1 needs to be small enough to light the lamp, but not too small or the transformer will be stressed when the button is pushed. See if you can see the rating of the doorbell transformer, they are often mounted directly on the main breaker box. R1 should be sized (wattage) so it doesn't burn up if the switch is held closed.

D3, or several diodes in series, or a zener, is optional, if needed to raise the threshold so the opto isn't on when the switch is open. I am assuming that 60 ohms is the cold resistance of the lamp. The hot resistance should be high enough to keep the opto off.







simulate this circuit &ndash; Schematic created using CircuitLab