Have a new, custom "post notice" created for your site, rather than the much larger development of a new actual delete reason.
This, effectively, already exists, but in a different form than you've requested. Moderators can add a "post notice" to any post. I believe the default ones are "Needs citation", "Current event", and "Needs detailed answers". Post notices for a site can be edited and/or added by Stack Overflow staff. I've considered requesting for a "Deleted due to being AI generated content" one to be added on Stack Overflow, but have not, yet, done so.
Adding such a notice is a separate step for the moderators, rather than being directly integrated with deleting a post as you've requested. Having the underlying mechanism of what you're wanting be a post notice is substantially more likely to be implemented, as it's something that already exists within the system, rather than needing to develop a completely new "delete" methodology and burdening every delete action with making a choice between type of delete being performed. If moderators desire, a userscript could be used to have a single button for "delete and add AI post notice", but the usefulness of that will depend on the volume of posts where the post notice should be applied.
How to get a new, custom post notice for your site?
I'd expect the process for requesting a new post notice to added to your site would follow the normal process for requesting such customizations for a site. Usually, this is done by creating a post on your site's child meta (i.e. not here on Meta Stack Exchange) which defines the wording to be used for the post notice and requests the change. If there's support from the community, or at least from the moderators, then the meta post can have a status-review tag added to it by a moderator to escalate the request to the CMs.