I believe after outbound travel BA can change the ticket , they have done this for me before with an Expedia ticket I was getting a non sensible answer about changing

5am sounds weird though, what route is this ? If it's Europe then another option may be a same day change if it's a Plus fare

lastly you may be able to just contact the corporate agent yourself and try ? We use Amex and I can just call them to deal with these post booking matters myself