No. The HSReplay.net replay viewer and infrastructure is optimized around "normal" , where replays are often a magnitude shorter than in Battlegrounds. As such, there are no replay pages accessible and Battlegrounds will not appear on your My Replays page.

This is especially due to the long and complex combats that happen in Battlegrounds. To put it into perspective: In later turns, just a single combat can easily have more attacks between minions that happen in an average Hearthstone match. This causes replays to be too long, which causes excessive loading times and costs.