Is there any improper or awkward usage in this sentence?

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.

This grammatical but it sounds stilted. If it were me, I'd be more likely to say: Dear Mom, I wish you could come to the U.S. for a visit. It would be wonderful to see you, no matter how long you could manage to stay. I love you and I miss seeing and talking with you.

"Satisfying" in English isn't really right for a mother -- I'd say "love." "Filial" is quite old-fashioned but you understand it. "Love" covers that too.

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."

Thus

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.

First, there is nothing inherently wrong with your original quote. It is entirely grammatical, and its meaning is clear.

Second, to help you with the word filial (M-W link), it is an adjective connoting something a child has toward a parent. The expressions filial love and filial obligation are not uncommon, referring respectively to the affection a son or daughter has for a mother or father; and to those tasks a child must perform for a parent, from a youngster doing the dishes to an adult child acting as caregiver to an elderly disabled parent. That said, whether "carry out my filial duty" is better than "carry out a son's duty" is a matter of opinion.

Finally, is there any part of your original quote that could be changed to improve it, including alternatives to the word "satisfying?" Again, I think that part of your question is most opinion-based. There are always hundreds of ways to say the same thing, all equally valid.

If you say you wish your mother COULD come to the U.S., it implies that she can't come. Thus, you need to say, I wish you WOULD come, as this implies that she needs convincing, which is the message I received from what you wrote. In writing, brevity and lucidity should be your goal and when writing to someone you care about, directness about your feelings. If your mother needs convincing to come to visit, I suggest the following: Dear Mom, I wish so much that you would come to the U.S so we could spend some time together. Even if you can come for only a short time, I will be happy just to see you again.