Cut and reglue the batten inside the two leaves - probably cannibalise the original softwood 'bottom rail', or use something that will serve your fixing purpose, and tuck it back in with a good dab of polyeurethane or somesuch grab adhesive...glue the inside of the door, not the rail, or there will be goo everywhere. As Martin says above, cut the top instead - but I would still finish it, not leave it open. Leave too much waffle exposed and your doors will look like big panels of SC&P.
Ooo - look what I found. all you need is a 400 acre smallholding, the contents of a Screwfix catalogue, a slow-motion body (No, not you, slo-mo) slow delivery and tons of spare time. And Jennie's garage.