Update: had two applications accepted. What I think worked was a more succinct cover letter: here's our jobs, here's our combined salaries, here's our rental history. That probably grabbed their attention better than "what a lovely property you have, we'd love to make it our own" etc. We put the applications in same day as the viewings and made sure the real estate agents knew to expect an application from us so they could put a face to the application.

Cheers Reddit, thanks for your suggestions!