Does the prayer count as having been offered outside its prescribed time without a valid excuse if a woman purifies herself after midnight and delays performing ghusl until shortly before Fajr, then the call to Fajr prayer is made just as she is saying the takbir for the 'Isha prayer? And does this entail kufr, and what is required of her?

It is not permissible to delay the prayer beyond its appointed time, as Allah says: "Indeed, prayer has been decreed upon the believers a timed ordinance." (Quran 4:103). Deliberately delaying it is among the major sins, as Allah says: "So woe to those who pray, [but] who are heedless of their prayer" (Quran 107:4-5), meaning they delay it beyond its time. What is obligatory is repentance, seeking forgiveness, and observing the prayer at its proper time. A menstruating woman, if she becomes pure before dawn, must pray Maghrib and 'Isha. And if she becomes pure before sunset, she must pray Dhuhr and 'Asr.