You should probably be supporting with air more than you are. I, with the help of my new teacher, am trying to move from killing my face (not lips, my face), to using the bigger muscles, which should do the real work.
As to whether the tiredness should be spread out between your top and bottom lip, I don't think it matters, depending on your style, but you should be taking the brunt of the strain on the muscles around your mouth which support your embouchure, not form it. If you are using too much pressure, you will feel extreme fatigue in your lips, and forming an embouchure later in the day will be painful, a way that I like to test this, is by swapping from my trumpet to my cornet, if the cornet feels uncomfortable on my lips, I was using too much pressure.
Do you have a teacher?