OK, I understand a bit better now, thanks!
I did not do the calcs on paper, but I 'guesstimate' that with an H7 you should manage up to 40Mbit or so. If that is enough, my approach is FreeRTOS + LWIP, where I store incoming data into a StreamBuffer (FreeRTOS' FIFO) and read out with the TCP Sender function. Unfortunately from in-buffer to out-buffer there is data copy involved. That has allowed me to stream 96KHz audio without glitches. You will need to pre-define your data structure to send all fast and slow data in the same stream, plus add some checksums for good measure.
My other approach would be, if you need GBit, to do the jump to a Linux system, with the ADC data being sampled in the FPGA, and stored into memory via DMA or any other methods FPGA's use these days for data transfer. Example, Intel/Altera has a Scatter Gather DMA which lets you grab data from a source, specify the RAM write address, and then the blocks take care of the rest pretty much. I don't know about Xilinx, haven't explored it. I would be surprised if they didn't!
Cyclone V offers dual ARM cores with FPGA fabric, but I would not do bare metal on them, let the OS take care of everything.
If you have never built an embedded Linux system, it is a relatively steep curve. Of course, manufacturers offer SD card images ready, but when you want to customize things, that's when it gets tricky. Prepare to spend one or two weeks depending on your skills getting your new image from scratch. And the time it takes to compile.
If price is a concern, you can't beat STM32 H7 dev kit for a few bucks. But you're limited to lower bandwidths.
Terasic / Digilent and other offer a range of boards with ARM hard cores + FPGA, but prices are usually in the 200+ USD or so region I believe. More for more powerful devices.
Not being too helpful, sorry!, but I'd be interested to see what you do next. Data acquisition over Ethernet is a subject I'm very interested in and have looked at in the past one way or the other.
Cheers,
Alberto