@jnelson well use your existing networks as the transit. And move your actual network to something else 192.168.5/24 on one side and 192.168.2/24 on other for example..
If you have no control over what network they use. Problem is you might have routing problems on their devices. But why can you not get with who manages the mpls routers to fix the problem.
Your going to have issues when device sends their syn,ack back to their gateway (pfsense) and pfsense never saw the syn to open the state.
The correct setup for what you have there is with transit networks.
As to routing on hosts, yeah you would need a route on the client that says hey you want to talk to 192.168.1/24 send it to mpls router at 4.1 vs pfsense at 4.254.
And on the other end your app server would need route to 4/24 to send it to 1.254 vs pfsense 1.1 address.
Problem with such a setup is you loose firewall between your networks.. I would really suggest you get with who manages the mpls to correct the setup. They should have no problems changing the ips to some new transit networks and fixing the routing. If they are currently using your pfsense IPs on each end as default.. Then you could change your networks and use the existing 4/24 and 1/24 networks as the transit networks.