I have an EL Fairlane.

Two things NOT working are the rear demist AND the power antenna.

The antenna is half raised and I have replaced the factory radio with an aftermarket, so I assume the antenna will no longer raise/lower.

The rear demist is needed to program new keys.

Is there a solution and could the BEM be faulty?

I checked both switches and they appear to operate by testing with a multimeter

I have an EF, and am assuming its the same.....

First make sure the antenna isn't seized. I had to spray mine with some WD40 and slowly work it until I could free all the sections. Some had bound and would not allow the antenna to raise.

As long as your stereo has a power antenna output then you can use that to enable the auto feature. My one loss with my aftermarket stereo is that it always raises even if I'm not using the radio.

The switch on the dash (if you have one) should change the antenna between half & full height when it is raised.

make sure also that you haven't plugged the demister switch into the antenna plug and vice versa. the plugs are very similar but thankfully the pin layout is different.