Unless you have an exokernel operating system.

Nah, that's not even necessary, I believe: the congestion window is a TCP feature, but any normal user space program (with a few privileges) can create IP datagrams with custom content, so you would only need to implement a bit of tcp in a normal old style monolithic kernel

And since you control the only higher level protocol (your http implementation) you can probably cut some corners and get done quickly :)

You may be able to create them, but as a server, how would you receive incoming connections?

No need for an exokernel here. A microkernel could enable the same thing, since your network stack would be Yet Another Service (TM). This could easily be implemented on the likes of L4.

I don't really see the difference between patching the kernel and patching the privileged network stack server in a microkernel environment.

In an exokernel environment, the network stack server wouldn't be privileged.