"Student body president" does not necessarily need to be capitalized because it is not a pronoun or a name of something that needs to be capitalized.
For example:
"The student body president has arrived." is more in check than "The Student Body President has arrived." which just gives the sentence that weird feeling and seems like it isn't English anymore.
As for "President", there are times when it is capitalized and not. It is only capitalized when it is referred to as a name. For example, "The president just left." And "I don't think it will work, Mr. President."