You definitely need the
gain, because otherwise you are implying that you are currently living in Panama.
If you want to omit one of the uses of
will in your sentence, you could replace the second one with
The experience I will gain living in Panama can assist me in understanding how Central Americans live.
However, that would slightly change the meaning of the sentence, so you might want to just keep the two uses of
Because it appears that you are applying for some position (possibly a Fulbright), you may be better to be less assuming and use
could instead of the first instance of
The experience I could gain living in Panama would assist me in understanding how Central Americans live.
On a side note, it may be better to be more exact in the last half of your sentence, and mention what exactly you want to know about the Central Americans. As you are probably applying for something like a Fulbright, the readers all ready know that living in this country will help you learn about the people. This means you are wasting half of a sentence that could be used talking about what you specifically want to learn about the people.