How can I arrange to do my own qualification of non-fully-qualified domains, and then pass them on to the next router?

If you have some list of domains that you want to qualify, you can do this using a redirect router. For example,

qualify:
  driver = redirect
  domains = *.a.b
  data = ${quote:$local_part}@$domain.c.com

This adds .c.com to any domain that matches *.a.b. If you want to do this in conjunction with a dnslookup router, the widen_domains option of that router may be another way of achieving what you want.