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.

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 :)