OK, I have figured this out, so I am sharing here in the hopes it helps others. I was doing a few significant things wrong.
A much-simplified Example:
New Site Network:
Netgate/pfSense firewall/router
192.168.1.0/24 LAN
Actually VLAN'd, half dozen subnets, etc.
Old Site Network
Cisco Firepower firewall/router
172.31.0.0/16
No VLANs, subnets, etc.
I needed to make the old network play nice with the new one until resources could be fully migrated over time.
I tried defining an interface on the new network that used an IP address from the old network, setting up routing and rules between the two, etc. That ended badly.
Maybe I had it backwards: I tried defining an interface on the old network that used an IP address from the new network, setting up routing and rules between the two, etc. That ended badly.
I tried setting up a new simple, dedicated subnet solely for the purpose of interconnecting the two routers to manage transferring data between the two networks, static routes, etc. That did work.
I call it a transport network, but I bet you networking guys who know what you are doing actually already have a name for it (I'd be curious what that is).
Where making the new network at one geographic site talk to the old network at a different site is concerned, I discovered that adding P2s on the IPSEC S2S tunnels was the trick (and not setting static routes, which I had tried).
Problem solved. Thanks.