Following your example of the CEO of LGB Alliance, the equivalent would presumably the CEO of Stonewall. Going with the occupational exemptions, it seems likely that the CEO would be expected to hold specific views which align with that organisation. Lower level employees would presumably not be held to this.
But,
From this website - www.jmw.co.uk/services-for-you/employment-law/blog/what-does-decision-forstater-v-cgd-mean-employers
'What does this mean for employers?
This decision gives gender-critical beliefs the same legal protections as religious, environmental and ethical veganism philosophical beliefs. Therefore, anyone sharing these protected characteristic are protected from unlawful discrimination and harassment.
However the EAT stressed that the judgment did not mean individuals with gender-critical beliefs could “misgender trans persons with impunity” and everyone will continue to be beholden to the prohibitions on discrimination and harassment within the meaning of the Equality Act.
It is important for employers to be mindful of the kind of workplace culture they want to promote. They will need to strike a fair balance between allowing freedom of speech and tolerating opposing beliefs and ensuring employees have a safe environment that is free from discrimination and harassment. Failure to do so, could result in the employer being liable for a potential claim either way.
As this case demonstrates that employees are generally entitled to hold gender-critical beliefs, it is important for employers to focus on thewayin which employees may potentially manifest those views, so if an employee is deliberately upsetting or inciting other employees, regardless of whether that belief is protected, it does not necessarily mean that an employer should tolerate it, or do nothing.
It is therefore important for employers to review their policies and handbooks and provide training and relevant education programmes, in order to help protect both the employees and themselves.
Policies and handbooks should include wording that managers and supervisors will not treat any employee unfairly because of their beliefs and that other employees must be respectful of other people’s beliefs. Further a zero tolerance approach should be adopted when it comes to employee’s promoting their beliefs in the workplace in a way that upsets or offends colleagues which could amount to harassment, and in turn could lead to disciplinary action being taken.
Ultimately as a result of this ruling, employers need to emphasise within their workforce, the important of tolerance and mutual respect in order to promote an open, equal, diverse and inclusive environment for people to work in.'
Maya Forstater made comments on social media, on an account which was not associated with her employers. She was reported to her employers by colleagues who believed that she should not hold these beliefs. It was noted that within her employment, she respected preferred pronouns and was generally respectful of transpeople. She just didn't believe that TWAW. There was discussion within the EAT about plurality of opinions within society and that even if someone holds an opinion, they are still expected not to create a hostile work environment.
The thing is, this applies equally to both those who hold GC views and those who believe TWAW. So if an employee of LGB Alliance believed that TWAW, they would be entirely entitled to hold that view but would not be entitled to continuely refer to their workmates as bigots or 'TERFS' for holding GC views, or to try to hound them out of employment.
But this kind of situation is happening. Women are being hounded out of employment because they hold views (which are legal and 'worthy of respect') which other people object to them holding, no matter how these views are expressed.