Linux

There are a couple of things that will get in your way with this.

Bandwidth

Let's go with the bare minimum of your high end given what you want:

• running both of your displays at 4k 30Hz 8bit only will require 6.66Gbps per display
• 2.5Gbps networking is self explanatory
• assuming you only want USB 2.0 ports, 480Mbps per port

without overhead, that's ~17Gbps. USB 3.2 Gen 2 can do 10Gbps, and USB 4 can do 20-40Gbps, so it would need to be a USB 4 dock at minimum, which means new and most likely above your budget. Your low end could probably be done on USB3.2 Gen 2, but you're still going to come close to your budget or blow it.

Multiple displays

Running multiple displays from a single usb-c port is not great. you can do it with thunderbolt docks just fine, but they are all going to blow your budget. With usb-c your options are a single display per port on your machine with displayport-over-usb-c implemented, or multiple displays using multi-stream transport (MST). MST is known to be extremely finicky and generally not worth the hassle in my opinion.

Recommendation

If you need multiple displays (on top of the HDMI 2.1 port on your machine), either dedicate both usb-c ports to it and use two cheaper docks, or go all in and get a thunderbolt dock like the Caldigit TS4.