I think if this was the case, this site, and other social medias, would've been exploding with talk of the inauguration being shown ahead of schedule.

Also, let's be real, the DS would not make a mistake as rudimentary and as disastrous as showing the pre-recorded inauguration to other countries before America.

If they went to the effort of doing a pre-recorded inauguration, they will have sent it out at the correct times, in my opinion.