Unfortunately the CA/B Forum has high requirements for constrained subordinate CA certificates[1], which to me sounds a lot like regulatory capture.
[1] https://community.letsencrypt.org/t/sub-ca-with-wildcard-cer...
[1] https://community.letsencrypt.org/t/sub-ca-with-wildcard-cer...
So that's not going to be free. But it might be possible to do it if you were big enough to pay for it. I have dreams of having my private CA also signed off on by webpki so apps and browsers could use the same servers without having to include webpki in my apps.
Not that I really work on such things anymore.