Sure, people do that all the time. After they run "InstallRoot" to install DoD root certs on their COTS device, that is. I'm honestly not sure any major browser will allow you to use a client smartcard without having the smartcard's certificate chain to the trust store used by the browser so this part seems unavoidable.
FWIW I just tested it and yes you can run a web server using a commercial server cert that enforces client PKI tied to the client having a DoD PKI cert. It works just fine.
> I'm honestly not sure any major browser will allow you to use a client smartcard without having the smartcard's certificate chain to the trust store used by the browser so this part seems unavoidable.
It’s been a while, but I’ve used file-backed client certs issued by a private CA in an ordinary browser without installing anything into the trust store, and it worked fine. I don’t see why a client cert using PKCS11 or any other store would work any differently. Why would the browser want to verify a certificate chain at all?