Two issues I've noticed:

1. Clicking 'challenge "NPC" will not let me start the fight. When I click start battle, it just refreshes the preparation page over and over. I have to manually go and find the opponent from the listing to actually start the fight.
2. I'm getting NPCs I apparently don't have? ...When I do have them. I know I have the opponent because I've fought them before, but they aren't showing up in any tabs and when I try and fight them from the challenge button on the quest page, issue 1 happens.

I know, I can just skip this, but it's been happening repeatedly and it's super annoying.

Once you get to the "Start Battle" page, double check you actually have a pet selected with one of the radio buttons.

By default it tries to select your "active" pet so if your active pet is not eligible for battle (Fed/Happy/Has health) it could just be defaulting to not having a battle pet selected so it refreshes the page.

The Battle Quests seem to be selecting and working normal for me otherwise!

example

iirc BQs draw from a pool of opponents, regardless of if you have them or not(as many are from old events etc)