On doing further research, it looks like this site's webhost won't have enough memory to run Discourse without paying a higher web hosting fee each month. Given the low activity here, I don't think it's worth it. So, I'm leaning towards a minimalist custom forum that is an extension of the main site. Something like the Ghost Support forum is what I have in mind:
Simple and clean. I'd add a few extra things such as Neil's An'jou emoticons, but keep it roughly as minimalist.
It should be able to port the old accounts and posts over too; I'll make that a priority, since resetting the accounts would probably kill the forums at this stage!
Are the Ghost forums linked above missing any critical features? Maybe clearer read/unread messages, but otherwise they seem good to me.