That much headspace isn't going to cause a problem. It's the spigot you need to watch out for. Many people have learned the hard way that those bottling buckets are designed for bottling, not for long term storage. If that's the way you want to go, then good luck with it. Just be sure to keep some towels on hand, or have it near a drain.
Personally, I wouldn't risk it. I'd use the 6 gallon carboy for my primary (I'm assuming you have a blowoff tube) and either buy a 5 gallon or just forget about secondary.