I just used a cheap one off Amazon -

https://www.amazon.com/NewerTech-Headless-Video-Accelerator-Dummy/dp/B01ASJCZFK

Worst case if it doesn’t work out it’s easy to return. Full disclosure I only used this for 1080p but from what I’ve seen it will support up to 4k

How to turn a Chromebook into a Linux Remote Desktop:

1. Enable Linux (crostini)
2. Install LXDE (or any other Desktop Environment of choice)
3. Install Chrome in Linux
4. Install Chrome Remote Desktop in Linux
5. Add Stay Awake extension in ChromeOS browser to keep Chromebook awake: https://chrome.google.com/webstore/detail/keep-awake/bijihlabcfdnabacffofojgmehjdielb
6. Add virtual external monitor
   1. USB-C to HDMI dongle: https://www.amazon.com/gp/product/B07FT9Y9H9
   2. Virtual 4K HDMI dongle: https://www.amazon.com/NewerTech-Headless-Video-Accelerator-Dummy/dp/B01ASJCZFK
7. Close chromebook lid with a power supply attached.
8. Remote into your new Linux machine and never open the Chromebook lid again!

Bonus: you can reboot Linux and CRD will be back and connectable after the reboot without having to open the Chromebook.

Technical note: the virtual external monitor is required for the Chromebook to stay awake with the lid closed.