I am a Mexican citizen in Mexico on the US border and cross almost weekly, so I am legally able to visit. My passport is currently expired and it takes some time to renew it. I am afraid TSA might not be able to except my B1/B2 even though it is an American issued form of ID. I have been looking around the internet and haven't been able to find a similar situation. I have a valid I-94 form as well that grants me permission to travel pass the 100 or miles allowed in the state I travel in.
The list of TSA-accepted ID includes "Border crossing card," so that should be considered acceptable for airport security purposes.
However, keep in mind that you can't use the card alone to cross the border by air, so that will limit your travel options and prevent you from flying back to Mexico if a family emergency or something requires a quick trip home. Unlike a passport, it's also not an incredibly widely known form of ID in the US, so hotels and other businesses that require ID may not be willing to accept it, especially away from the border. You could contact them in advance if you're not sure it will be accepted.