This is what happens when you use a commercial piece of equipment.
Your problem is the device you call a router from say starlink is not actually a router. It is best called a gateway its function is to translate a lan subnet to a single wan IP address.
Your switch actually can function as a actual router. A router does not translate addresses it purely moves traffic from 1 subnet to another without changing the IP addresses. So you could have say a bunch of subnet 192.168.10.x 192.168.11.x 192.168.12.x These device all would have a gateway ip that is the switch but on different vlans. The switch would then allow these device to talk to each other using those IP without the need to change them.
Your problem is you need a device to translate all the 192.168.2.x ip addresses to something in the starlink lan network of 192.168.1.x. Now maybe, but unlikely, the starlink has the ability to route the 192.168.2.0/24 subnet to some interface on your switch which would then send it to the proper device. This generally is not a function a consumer box has but I have seen it on some.
What you need is a consumer router to sit between the 2 networks and translate the IP addresses.
I am not sure I agree with your assessment, so don't understand what you're proposing.
The issue is that the Starlink router has a DHCP server and it can't be turned off.
I had the Cisco switch on 192.168.1.x but it was causing issues, as the Starlink Router was also assigning 192.168.1.x addresses.
So what "consumer router" do I need to sit between the two?
Because right now I have a TP-Link E705x sitting between the Starlink router and the Cisco CBS350 -- I need to connect the 192.168.2.x network to the internet wirelessly. Unable to run a cable.