Given only one possible appointment for masters application interview with no possibility to renegotiate date: Is this a red flag or normal?
I think it was perfectly reasonable of you to ask to reschedule.  Of course, it's unfortunate that they weren't able or willing to accommodate you, but that's up to them.

You'll now have to decide what you want to do: interview while traveling, or alter your travel plans, or skip the interview altogether.  It might be worth considering whether an institution that seems so unwilling to accommodate prospective students is an institution that you'd want to attend at all!