Well, in my opinion the easiest way to do that is to run DNS off the firewall so the queries are organically sourced from something interesting to the tunnel.
In the Resolver (unbound) you can set one outgoing interface for queries it needs to resolve. That could be the LAN. All queries going to the outside will then have to go through NAT but that is generally not an issue.
In the Forwarder (dnsmasq) you can set the source address on a per-domain-override basis.
Again, if it important, I'd run a couple of bind instances on the inside.