"If he became president, what would that mean to ..."
The ideas could be expressed in a number of ways. It is acceptable as is.
The first phrase is a conditional clause and the second represents the results, if the condition is met, in the form of an inquiry. The condition can be in the past, as the results cannot occur until the condition is met.
If he becomes president....
If the condition is in the present, then the inquiry in this case should be:
what will that mean...?
If the condition is met in the present, then the results need to be beyond the present (the future).
In older common usage the wording might have been:
Should he become president, what would that mean...?
But such usage is rare today.
But I'd never use the conjugate the verb in the past when it's an something that might still come to pass.
It would be the order of events that determine tense, as a general rule, if a
conditional is used,and, a situation counter to fact is created. "If he became president" suggests he "could have" become president, but clearly means he is not president. It also means that if he were president, he became president before the present. That is OK, as the results of his being president
cannot be until he is president. But, whether he "became" or
becomes" this has not occurred, so the results are in a future time. Whether one starts in the past and goes forward, or begins now and goes forward, the relative time is correct. The use of a conditional suspends fact and time becomes relative, not factual.