No, there are no web APIs that an attacker could employ to modify the HTTP version string in a reflected XSS attack.
I often come across similar flaws during penetration testing that are technically XSS, but not directly exploitable without manually altering the HTTP request. These bugs often get rejected by bug bounty programs. Another common example for this type of flaw is the
You would often see that
vulnerable.example is reflected in the response body without any filtering but you simply can't choose the value of the host header in a real-life XSS attack, so the vulnerability turns out to be rather useless. Yet another example is XSS via the user agent. Similarly - unless it's a persistent XSS flaw, just printing back a modified user-agent string will most likely not be exploitable.
Beyond reflected XSS there are obviously other vulnerabilities to look out for. The response might be cached somewhere, enabling you to turn it into a persistent XSS flaw. Or the value might be stored unfiltered in some backend logging system.