Before I go and cobble together a bunch of scripts to drive dnsmasq and try to figure out how to tie them into NetworkManager: How are other people nicely handling access to home network resources and company resources both of which are via private DNS servers?
Ta. That looks like it's exactly what I want, but not obvious from the GNOME3 NM configuration dialog, and nm-connection-editor seems to have slightly different details. I shall have a poke later.
Adding "dns=dnsmasq" to the [main] section in NetworkManager.conf and adding the search domains to the VPN connection via nm-connection-editor has done exactly what I want. 2nd step wouldn't have been needed if my VPN provided the search domains (not possible with PPP AFAICT).