This all reminds me of the advice is Patrick McKenzie's blog post "Identity, Credit Reports and You." Specifically: you never want to deal with customer service, they are the Department of Fobbing People Off. You want to be communicating in writing with a lawyer, since lawyers have the power to tell other people in the business "you're creating liability, make this problem go away".

In this case I think you don't have the same structure as in the credit report case (an act that says they must investigate and respond within X days), so for credibility you probably do need a lawyer writing a letter for you. But I strongly suspect that something like the following will generate a response: "Dear Sir/Madam,

I represent [user]. My client is a customer of your business but is unable to access your services due to [disability]. He has communicated with your customer services (see attached screenshots) and requested they provide a means for him to access these services. Unfortunately they have declined to do so. [Relevant legislation] requires reasonable adjustments to be made in serving disabled customers, and my client and I believe that [adjustment] could easily be made by your business to allow customers with [disability] to access these services. Please advise us within [x days] what adjustments you plan to make to allow my client to access these services.

Yours,

[Lawyer]"