You should add a diode on each relay coil to reduce the reverse peak voltage that may ruin the switches on the first relay, and make them stick.
It may also be a good idea to add some sort of snubber circuit over the bell switch(es) if they are coil based. 47 ohms in series with 100nF should make a start value.
Regarding the problem with the LEDs in the switches, it is obvious that the current through the LEDs add up to the hold current for the relay, and makes it stick. To fix that you need to reduce the idle current through the relay. if the relay worked with one LED, you need to bleed the extra LEDs current away from the relay coil. A resistor of 100 ish ohm may be low enough to do this, but remember that this resistor will have 12V over it as long as you push the button and burn some 1.5 W. If you use 4 x 27ohm, 0.5W resistors in series you should at least have a safety margin on the wattage, even if they get hot.
It may also be a good idea to add some sort of snubber circuit over the bell switch(es) if they are coil based. 47 ohms in series with 100nF should make a start value.
Regarding the problem with the LEDs in the switches, it is obvious that the current through the LEDs add up to the hold current for the relay, and makes it stick. To fix that you need to reduce the idle current through the relay. if the relay worked with one LED, you need to bleed the extra LEDs current away from the relay coil. A resistor of 100 ish ohm may be low enough to do this, but remember that this resistor will have 12V over it as long as you push the button and burn some 1.5 W. If you use 4 x 27ohm, 0.5W resistors in series you should at least have a safety margin on the wattage, even if they get hot.