I've been through BUNAC but haven't had that problem.
Really like you said you can't talk to anyone on the weekend. I'd go out and get the money order this weekend, before you leave the country, and put it in the mail to go out on Monday. Then if possible, have someone call them Monday and explain the mistake and that the money order is in the mail. No worries; I'm sure it happens a lot! The worst that will happen is it will delay things a tiny bit. I don't see why they'd need to return the application to you.
Flight to Tokyo: 92,000 yen
Metro to Ginza shopping district: 160 yen
Watching middle-aged Japanese woman dance to U2: priceless