About the first question, I wouldn't expect that you will use any mixer very often based on what you described yourself. For the second question, if you really want one, get a German mixer for about 200 USD. This machine can only do mixing and it is good for making tough dough.
There is another solution. Ask for your hu*****and to make the dough for you. Then you make the filling. When putting everything together, you can ask for your hu*****and and your son join you together and make Baozi. With this solution, you are not alone when doing kitchen work and you don't need to buy anything which you don't need very often.