You are going to to need the Starlink ethernet adapter pretty much no matter what you do.

Ideally:

1. Run ethernet cable from starlink to a switch inside the house
2. Run ethernet cable from the switch in the house to an Access Point (AP) in the house
3. Run an ethernet cable from the switch in the house to an access point in the workshop.

Alternatively

1. Run ethernet from starlink to a point to point wireless bridge in the out building
2. The other side of the wireless bridge will be mounted on the house.
3. Run ethernet from the wireless bridge on the house to a switch in the house
4. Run ethernet cable from the switch in the house to an Access Point (AP) in the house
5. Run ethernet cable from the switch in the house to an Access Point (AP) in the workshop

Edit:

Switch: https://www.amazon.com/Ethernet-Splitter-Optimization-Unmanaged-TL-SG105/dp/B00A128S24/

Wireless bridge:https://www.amazon.com/NanoBeam-NBE-5AC-Gen2-US-airMAX-Dedicated-Management/dp/B07NNWY9Y8

Access Point: https://www.amazon.com/TP-Link-EAP225-V3-Wireless-Supports/dp/B0781YXFBT/