You are incorrect about mammograms. While men can get breast cancer, mammograms are generally not recommended for them because of the relatively low occurrence rate and that fact that they are designed for larger breast tissue mass. However, that may not apply to the transitioned population (both directions). It depends on a number of factors and should be discussed with a medical professional.

Some sauce for those that are curious:

https://transcare.ucsf.edu/guidelines/breast-cancer-women

Those aren't emergencies and it would come up in any prior appointment or while making the appointment. It may be a bureaucratic issue if insurance doesn't want to pay for a (diagnostic) procedure that's not in accordance with the gender on record. But that's something fixable with insurance policies becoming more inclusive and not something that would lead to worse health outcomes for the patient.