No problems, just follow the cable suggestion from "badgerAteMyHomework".
Start at the top attach a small weight to the cable, https://www.amazon.com/Monoprice-Cat6-Ethernet-Patch-Cable/dp/B01M3VDWKB and let gravity do the work of running it downstairs.
I would suggest that you get a TP-Link Deco Mesh. It can be configured in AP(Access Point) mode or in Router mode. So you would connect the first one to the Ethernet coming from the Starlink. Configure your Deco using the APP on your smart phone. Set it up in AP mode. The connect and configure the next Deco node. You can have up to 10 Deco satellite nodes in the mesh.
So once you have all of the Deco mesh nodes configured in AP mode, use the second port on the Main Deco Node to connect to an unmanaged Ethernet switch. From there connect all of your Ethernet lines to the switch.
Now connect the Ethernet cable in your detached workspace to one of the Deco nodes. If you need more than one Ethernet connected device in the detached workspace, just add another switch, either before the Deco in that space or after using the second Ethernet Port.
Now you will have Wi-Fi everywhere using the same SSID and multiple Ethernet connections.
The main thing to make sure is that you either place the Starlink Router into "Bridge Mode and disable the native Wi-Fi. OR Simply ignore the Starlink Wi-Fi and connect nothing to it.
I have had TR-Link Deco Wi-Fi Mesh nodes in my detached, un-heated, garage for many years, with no problems.
I live in N/W Ohio where it occasionally gets down to 0˚F for days on end and in the Summer it can get well over 100˚F.
The Eero 6 has Operating temperature range of; 0˚C- 40˚C / 32˚F - 104˚F. That is the exact same specs as my TP-Link Deco XE75Pro. You should have no problems.
Here's hoping that the distance and construction allows a good signal and connection to the nearest Eero node.
Power line is no replacement for a decent wired or even 6GHz backhaul