mattytrog released a guide for installing a trinket-type modchip/payload pusher inside the dock, if you want it automated. It's somewhere in the guides subforum. Edit: here
https://gbatemp.net/threads/modchip-dock-modification-with-samd21.521301/
If the built-in screen doesn't work but the console still boots when undocked, you can use any usual method of entering RCM and pushing a payload, but if it won't boot at all unless docked then a chip in the dock is probably your only option. No RCM payloads use the dock's video output, so you won't be able to use hekate's menus and tools etc, but using fusee-primary or hekate autoboot to start Atmosphere would be fine.