The TSA issue with photo in prior Swedish passport was almost certainly not related to gender identity.

It was almost certainly related to some other issue, perhaps including aspects connected to how the Swedish government was handling and processing passport photos at the time of her passport application.

If I were the OP, I would merely insist on using the LPR card as photo ID with the TSA. It might require escalation and re-escalation, but eventually someone should acknowledge it as acceptable form of photo ID -- whether or not they accept it as ID for the OP, who knows.