Whenever I really try to push my tongue against the roof of my mouth as far back as I can, I cannot breathe at all and have to let go of the position. Meaning whenever I breathe in I am not mewing correctly. What can I do to improve this situation?
Try doing nasal breathing exercises while placing the tongue this way, to identify at what point you can start breathing. But do try to keep the tongue as much as possible on the palate.
Here is a video with myofunctional therapy exercises, at 0:45 you can see the nasal breathing exercise where you alternate your breathing by pressing one nostril at a time:
Here is another one, from Yoga perspective. You dont have to sit like that if you dont want to:
I also recommend placing the index and heart finger between the eyebrows, when you do this exercise (we did it this way when I had yoga lessons years ago, and it helps increase awareness of the sinuses/breathing and air flow).