Thanks for all the replies and advice. I always try to keep 2 filters running on any tank, partly because I like to over-filter, and because I'm paranoid about a filter ever breaking down.
So I left the Eheim running on my old tank, but took the FX6 off and replaced it with a second-hand FX5 I picked up on Donedeal (Irish equivalent of craigslist). So handy that the FX5 and FX6 connectors are interchangeable!
I decided not to transfer any water, just putting new water (treated with Prime) into the new tank. Then I put the FX6, complete with all its media, onto the new tank, and backed it up with a $150 cheap Chinese canister filter. I also planted a few live plants in the new tank. Then I threw a school of 8 juvenile Giant Danios into the new tank figuring that they would be expendable in the cycling process but I'd still be very happy to keep them if they survived.
Over the next few days I tested both tanks for ammonia twice a day, but not a trace of the stuff. So the Eheim seems to have enough bacteria to keep the old tank balanced, and the FX6 with its media immediately cycled the new tank. So I transferred my blood parrots in as well, and they're doing fine. Still not even a hint of ammonia.
Asking for advice on these forums can be a bit confusing, since everyone has their own opinions, but thanks to everyone who posted (even those whose advice I ignored). The responses on this thread gave me the confidence to give it a go, while helping me know what to keep an eye out for. I'm very happy with the resulting setup.