I have suggested a compromise where they note (as they have done with the word "queer" earlier in the guidance) that whilst some people are identify as or are happy to be referred to as "cis" it should be used with caution when describing others as some people (particularly feminists or those with gender critical views) find it exclusive and / or offensive.
I suspect the problem you've run into is the fact that most corporate HR depts still accept Stonewall's advice as the gold standard for inclusion (and many are under a great deal of pressure to get a good Workplace Equality Index ranking). If they're involved with Stonewall, these inclusive guidance documents will be submitted to Stonewall for judgement as to how well they are following Stonewall's guidance.
Stonewall says that gender critical beliefs are akin to antisemitism, so any mention of gender critical beliefs is going to set hares running for any organisation with links to Stonewall. Which is most organisations nowadays.
They will either be reluctant to consider a GC perspective because people believe Stonewall when they say that GC beliefs are bigoted, or reluctant simply because they realise that not using the Stonewall-mandated terminology of 'cis' means they will be marked down in the Stonewall index and the people within the organisation who want to be seen to be more inclusive (through Stonewall rankings or otherwise) will be annoyed with them.
If the person responsible for the document makes that change to delete 'cis', they will eventually have to explain their reasoning. It doesn't sound like they understand it well enough themselves to be confident in explaining it, so they're understandable scared to make the change.
They may well have googled the term to get clarification on why this term that they thought was standard inclusive language could actually be seen as offensive, and may have stumbled on all sorts of misinformation online as there are plenty of really strong options out there. I can see why someone who has been told 'if in doubt, follow Stonewall's advice on inclusion', might be worried about deviating in a way that is criticised as bigoted by many people online.
Also, given that the person responsible for the document isn't really aware of the controversy and is probably only just realising they've stumbled into a wasp's nest, I doubt they feel confident in changing the wording in a way that they could confidently say won't offend anyone - they most probably realise they're out of their depth now, and what do people do when they feel out of their depth? They consult the experts. And who are the established 'experts' on inclusion? Stonewall, of course!
My advice would be to offer to re-work the sentence(s) for them, to make it as easy as possible to accept your suggestions. They need support if they're going to find a way to be truly inclusive, as they clearly don't appreciate the problem (or the misogyny).
I also think it's important to flag to them that presumably the document should be inclusive to everyone and not just trans people, otherwise that's not inclusion at all.
Listening to trans people is obviously necessary and commendable, but trans people only speak for themselves and not for all the people who haven't or don't identify as trans for whatever reason.
Personally, I'd steer clear of championing feminist/gender critical viewpoints by name, for the reasons mentioned above. I would instead focus on the fact it is misogynistic language and that Stonewall represent (some) LGBT+ people, but have never claimed to speak for women as a group. I would also point out that even other LGB organisations reject the term 'cis', so it's not even an established norm within the LGBT+ community as times are changing and this is outdated.
Furthermore, since the word 'cis' is recognised as controversial by many and is the subject of much debate (as evidenced by a quick goodie search) whereas there are plenty of other words that aren't controversial, why choose the controversial word for inclusion guidance which is specifically designed not only not to offend, but to support other people in not inadvertently offending with their language choices too. It undermines the whole purpose of the document.
I'd also point out that directing employees to throw the term 'cis' around at each other risks exposing your employer to liability for discrimination claims if employees do object to being labelled as 'cis' by other employees, as they will point to the fact the employer directed them to use this discriminatory language. You've warned them it's an offensive term, so if they choose not to heed your warning, it's that drafter's fault if the employer gets sued. Given that they are obviously trying to be more inclusive, a discrimination claim isn't going to look great.