For example, can I change the word 见面 into 见他的面 or 见帅面 or 见这面?
If they can write it in Revelations, must be ok.
Rev . 22:4 And they will see His face, and His name will be on their forehead.
However, there is a meaning shift from 见面 'meet' and 见他的面 'see his face'. 2. below would not work, which is what you are suggesting, I think.
In 见面, 面 usually doesn't mean 'face', and it's like a classifier, referring to 'times of meeting'.
So, we don't usually say 见他的面, 见帅面 or 见这面. 见他的面 might be ok, but better say 见他一面 or 见他（本）人. You could say: 见他一面(meet him one time)， 见一面(meet once)， 见几面(meet several times)， 见不到（一）面(can't meet even one time).