Dear Mom, I wish you can come to the U.S. and reunite with me. Even if it is just a short period, as long as I can be with you and carry out a son's duty, it would be satisfying.
You should use the present unreal conditional "could" instead of "can," because you are not currently together and therefore this is a statement contrary to fact (unreal).
Your mother is not going to reunite anything by herself. You and her will reunite. Again, using present conditional, you and her could be reunited. Note: This uses passive voice (generally frowned upon) but this turn of phrase is very common and idiomatic.
In the second sentence, "it" seems to be used in a few different ways. Is "it" the time period, or is it the reunification? I think it is the reunification, so it should be "Even if it is just for a short period."
"Period" implies (albeit weakly) something that repeats or is one of many (c.f. periodic, lunch period, Jurassic period). This visit will be unique, so I would simply use "time" instead.
"Son's duty" is not a phrase I see commonly. If you're looking for something a bit more idiomatic, you could try "be a good son."
Dear Mom, I wish you could come to the U.S. and we could be reunited. Even if it is just for a short time, as long as I can be with you and be a good son, it would be satisfying.