Standard response: こちらこそよろしくお願いします. This a good question. I also find responding to greetings and compliments in Japanese really difficult because I don't want to sound like a robot giving an automated response but it's hard to be creative if you can't determine what kind of responses are within the realm of normal. That's why I always makes sure to take the initiative, lol. I have learned natural responses for most situations by watching Japanese dramas, but seasons greetings in letters are still extremely foreign to me.

I would say 明けましておめでとうございます。こちらこそよろしくお願いします。