I would use should, because getting a father a present is proper, or can even be considered a duty(Who wouldn’t? ) Although you can use shall, I think should is better in this situation.Should:
Used to express the opinion that the action of the main verb is one of duty, or what is proper, or what is a good idea.
Used to express plans or intentions concerning the main verb, esp. with regard to the future.
In my opinion, there is no difference (in AE). This is a question. With either "shall" or "should", the question means "What is your opinion about the gift I will choose?"My father’s birthday is coming. What shall I get him?
Is there any difference if I use "should" instead?