Ned, I suggest you apply anyway within those airlines you fancy.
The EU Passport request is in most cases just a sorting tool, as a work permit issued in any EU country is legally sufficient for a piloting job, (even for work in another EU state.)
If you have an approved license and some good and useful aviation experience, that should be a reason for recruiters to look past your passport
A language skills request might however be a bit more problematic to get around if you donīt speak the language.
I myself come from a country outside of EU, trained in the UK and Arizona, and I am now working for a danish airline. No problems whatsoever.
If you choose to take time off flying, you could use it studying something aviation related or working in operations departments for an airline. It sometimes proves out to be a shortcut into the right-hand seat when the time comes.
Good luck and welcome to DK..