I feel like Docker doesn’t give you a whole lot of help on doing this: to really make it work with Docker, you need to know enough about Linux networking in general that you don’t need Docker to set it up. That’s especially true when both isolated networks have their own firewalls.
If you already understand IP routing, what a router is, where a router would fit in this diagram, how you’d build a router on Linux, how these networks connect together, what DNS is, what a DNS MX record is, and how you’d configure the various routers and firewalls…yes, you probably could use Docker as a base layer. If you don’t already have that footing, adding Docker as an additional component to the mix and immediately needing to break its standard networking setup won’t make your task any easier.