Help with Multiple Router and hardwire set up

[drw930]

Just got starlink residential v2 and set it up. I think the connection is good so want to continue the full set up for my house. 
 

I have the dish set up outside on my barn. The router is inside the barn. That’s where my office and game set up is. Only thing out there. Preferably want hardwired for better gaming connectivity. 
 

There is an Ethernet running underground into my house that connects the inside router and Wi-Fi. 
 

I want to be able to set this starlink up as a hardwire for barn connection… then run the Ethernet to the house inside. 
 

Can someone please confirm if this setup below will work. Or if anyone has a better idea. The nodes will not work since the structures are about 85-95 ft away. 
 

Starlink dish > connected to starlink router. > using a starlink Ethernet adapter, I will plug an Ethernet into the adapter and attach a 4 port Linksys switch. > from there I will hardwire my PlayStation into the port switch in the barn. Then attach the underground Ethernet from inside router to the port switch in barn. 
 

Will this work? 
 

Thank you 

[RicochetStarlink]

It sounds like this should work just fine!