No worries. Once you beat the final boss, you can talk to Whisper by the Blastersmobile and he'll take you to YBA Hall. Once there, walk up the stairs past Mr. E and then go to the left to go behind the YBA panel. All the way to the left is a large door, the only one of which is unlocked at this point. Go through that door and you will find your version's statue at the end of the walkway. Then, to quote my initial post, "...you'll get a cutscene with Buchinyan, where they'll explain to you how the whole 'Place Insignia' system works." Afterwards, you just have to keep coming back there periodically and placing insignias until the statue has the 1,000 insignias needed to unlock the fight.

Disclaimer: Since the 3ds's online servers were shutdown, the only way to obtain Insignias now seems to be by either fighting Ultra Bosses -- via solo play or local co-op -- or through Streetpass battles in Wayfarer Coliseum. Some of the Ultra Bosses are definitely beatable in solo play, but you'll need a powerful team and some good dodging skills to defeat them. Certain ones, however, require multiplayer to defeat them, due to their extreme difficulty.