The "cervix-haver" label doesn't really make any difference to transwomen - no amount of linguistic adjustment will give them a cervix.
Of course it makes a huge difference to them. If "woman" is understood to be the sex designator word for female adults of the human species, no different from vixen, cow, doe, hen or sow, then they cannot be women.
So first of all, the word "woman" must be separated from female biology. So it's not a campaign for women to take up their invitation for cervical screening, it's an invitation to cervix-havers instead.
But it makes services involving cervices (I like that!) more inclusive for transmen.
It doesn't. Female transgender people face a whole range of barriers in accessing cervical screening. Healthcare campaigns about cervical cancer being framed as a women's health issue is a tiny problem on that list, and changing the language to cervix-havers so they feel more included is a shallow and inconsequential solution. Inconsequential for them.
However, we have large numbers of women in hard-to-reach groups that are put at higher risk of cancer by using that language. As cervical cancer charities, both national and international, keep emphasising, between 40% and 50% of women do not know what a cervix is, what it does and that they have one. So if you're talking to "cervix-havers", at best you're reaching only half your target group. Uptake of invitations has been falling for ten years and we cannot remedy that by no longer talking to the target group in language that communicates clearly.