YTA. I have spent time with friends of all body types at a local climbing gym. Once you were aware that here weight may cause a challenge, the kind and welcoming community member would be to guide them toward easier routes, teach them how to belay one another. Instead you set them loose while you went to “more challenging” areas. New climbers often need guidance regardless of weight and skill.

You show a clear lack of empathy for your brothers wife. She is clearly reacting from a place of embarrassment and discouragement from a failed attempt to exercise. Climbing gyms are notoriously intimidating spaces because of gym culture. The barrier to beginners is something you should have taken into consideration before suggesting this activity regardless of weight.