Computer's Static Routes being overridden by Switch Gateway

I have two buildings connected with fiber L3 switched using different subnets.

Building 1
192.168.0.0/24

Building 2
192.168.1.0/24

I have a 3rd party router at Building 2 that I need to tunnel out through a specific LAN IP its using, 192.168.1.150. So I set a static route on individual PCs at Building 2 so that when they try to go to 10.10.10.10 (example IP) it routes them to 192.168.1.150, everything is working good there.

Now Building 1, that same route does not work. We can ping all devices, even the 192.168.1.150 address, however if I set the same static route to the 192.168.1.150 address it doesn't reach its destination. A Tracert shows it hitting my Building 1 core-switch, then being forwarded to our Firewall, then out to the internet looking for the 10.10.10.10 address. So basically it knows it should be going across the fiber but once it gets to my switch the switch thinks it should be going out to the internet.

I can solve the problem by adding the same routing on the switches but ideally I want this route only to be in place for individuals we've added the Route for.

Does anyone have any suggestions?

Is there some way to program in multi-level static routes to guide it through each hop across the switches?

Thanks!