A fan would either do nothing (if the house was well sealed) or it would would draw warm, humid air in from outside and overwork the already inadequate AC system. You need to circulate air within the house or upgrade the AC system.

You should first have a professional look at the fan and duct system to see why it isn't performing well. Maybe it is just a bad filter, a blocked duct, bad fan, inadequately sized supply and return ducts upstairs.

We have a whole house fan that pulls air into our attic from the rest of our house. Not sure how it would work in your house with no attic but one of the companys that sells them could tell you. HOwever what we do is run it on cool nights then shut the house (or you could just do the upstairs) down during the day. The key is to strategically open windows, if there is one open right next to the fan most the air will come from there. You want the ones furthest from the fan that will pull air accross all the spaces that you want cooled. Because we are well insulated it works very well. But you need nights that are much cooler than the days for it to work.

Your upstairs sounds small, could you put a regular (but good)fan blowing inward in a window at one end of the upstairs and another blowing out at another end? Run at night and close windows during day. If you are using air downstairs you'll want to close off from the downstairs also. We did this type of thing before the wh fan and it worked pretty well (with cool nights).
good luck
kathy