I assume that if the file is generated, at least the directory it's generating to doesn't seem to have a permissions issue. The only other thing that stands out is the Vue message (in #3) about the formExclusions "setting a non-existent path".
Are there any pages added into the excluded page list for sitemap that don't really exist? Are there any pages entered at all - whether they exist or not - that you could temporarily remove and see if it generates correctly with them gone?
Does the site have a valid SSL cert? If it doesn't, then try to un-check the "Verify SSL" tick-box. If there's no SSL certificate, then that could be interfering with you having valid paths. The generator may have issues if it can't pass any validation tests when there is no such page as, (i.e.): https//mysite.com/blog/hello-world
You might also try to un-tick the box for "Allow redirects" to see if it generates successfully. If it works, then you might have directives that need to be modified for the site as a whole.
In my case, I have no excluded pages, the site does use a LetsEncrypt cert that is valid, and I've added a directive into my htaccess file to force all page requests to use SSL, so there is only that one protocol that would ever be valid. There are some other customized directives for redirects in htaccess, but they are security related and don't modify any routes generated by PageKit.
If none of those suggestions are of any help, I'm not familiar enough with Vue or PageKit routing to help with anything deeper.