I haven’t GM’d any burning wheel games yet - including mouse guard, until yesterday’s session. We had a veteran RPG-er (played big games like DnD, Hero, as well as some indie like Fiasco, Dungeon World), someone brand-new to RPGs, and a 9 year old girl.
The session went well, but BIGs was the most difficult part, and I couldn’t find lots of decent examples in the books (or even online.) Rather than stall the game out, we took a stab and moved on, here’s what we came up with. I’d love feedback and thoughts!
Caley - herbalist, healer, foolish, bold - ended up playing sort of like burnout druggie type.
Belief - Since there is no way to live after you die, why not live now?
Instinct - Always try and keep the peace.
This will be a difficult believe to challenge. I suppose I could try and play preparation and foresight against the moment. Staying up late, drinking/partying, hitting the group with obstacles that he doesn’t have the tools to deal with?
His instinct could come into play during any argument, conflict, or potential fight (physical or otherwise.)
Rose - solid fighter, brave, fearless - the 9 year old, she was hesitant, so her personality didn’t get to build. I hope to ease
Belief - Bravery is being the only one who knows you’re afraid.
Instinct - Always defend my friends.
We tried to help steer Rose, but it was difficult as she didn’t feel comfortable, and she didn’t develop a personality. I suppose putting Rose into positions where she SHOULD be scared, or SHOULD want to flee could work. The danger is that I’m unsure how a younger player will handle that. “Two hawks? Pfft, I’m brave, I draw my sword!”
Rose’s instinct can be called into play any time anyone else is threatened or in a conflict. (I see a pattern here) 
Pitting Rose’s fight tendencies against Caley’s flight tendencies would work too.
Whiskers - Fearless, clever, generous - the leader of the patrol, pro-mouse guard through and through, new to RPGs
Belief - Every mouse shall stand together for the good of the order.
Instinct - Always document the journey.
I think this belief is decent. It’s able to be challenged (separate the party, the towns, the guard, etc - physically or on issues like morality), and it is wide enough to do it on many levels - the party, the town, the territories, etc.
This instinct feels correct, though doesn’t (on first glance) have a ton of reasons NOT to do it. I originally suggested “Each night, I always document that day in my journal”, but it ended up like this. I suppose that situations where there’s no time to document could give Whiskers a choice, but it feels weak.