I'll probably get flamed for this, but here you go. If you can't get the cork out, push it in, and cork it again with a used wine cork. Like this
Notice the La Crema cork from my wife's wine, it fits perfectly. the original cork broke in half after being in a bag with a bunch of other stuff.
If you can't get the other cap out, push it in, re-cork it, and when you want to drink your bottle, pour it using a strainer. It's only a month, and if you are confident you and your people will finish that bottle, then it's - in my opinion, the best way to go.