I do empathise; the BMA are stuck between a rock and a hard place, in that their members want better work conditions, pay rises that actually match inflation (shock horror), but the government just say no.
The difference between the other unions and the BMA, though, is the BMA go back to their members and say "the government said no, sorry", whilst the others say "so be it", and the strikes commence.
I'm not quite sure how I'd improve the situation if I was a BMA representative; it's complex. However, there is literally no reason currently to pay the membership fee.
That's the sort of angle I'd go for. Organise coding camps and conferences, provide guidance on professional development, establish an online library, promote certifications. Make it a useful resource, not just a campaigning outfit. To get traction it needs to have a credible claim to provide value to employers as well as employees.
If an employer does respect their employees, values them and wants to attract valuable talent, they should see this organisation's activities as complementing their own goals more than they clash with them. That's the sweet spot to aim for. It is a tricky thing to pull off though.