Hi,
I don't know much about Employers First, but it doesn't sound independent to me. It's clear why the managers in your organisation want you to use them. Perhaps you need to look for someone who can work with both sides to ensure all parties are heard and have a chance to present their case. You could also contact your local
OHS authority if the issues are not resolved internally.
I was bullied by a woman manager in a previous job and there was no internal support for me because the woman (who only has entry level skills in old school
HR admin) had length of service and was good at working with the clients (adults with moderate to severe intellectual disabilities).
Unfortunately bullies are still considered acceptable in most organisations if they have technical, admin, or finance skills. Until this problem is dealt with on a national level and employers are forced to employ and retain people skills (because you can't teach empathy, compassion, and emotional self control) and train/coach/mentor staff in technical areas, you should do plenty of research before choosing a service provider.
Just my humble thoughts
Good luck
Cheers
Michaela