Why is the air conditioner running when the house is below the set temperature

I have read and re-read all the questions and answers every one put forth and here is another try at helping you resolve your A/C problem. You said that the temperature drop is about 10 degrees. You need to buy a pen style thermometer similar to a road pro RPCO 841 (amazon $11.00), and check the temperature drop at the furnace, close to the coil. If the temperature drop is actually 10-12 degrees, that is too low. A low temp. drop could be a worn out A/C unit, a dirty outside coil, low refrigerant charge, too much air flow at the furnace, a mismatched restrictor or thermostatic valve at the coil inlet, etc. I would check out as many as you can yourself, since I rarely trust any service tech. (I was one for 40 years). For the outdoor coil, turn off the A/C unit, look at the outdoor coil closely for blockage of dirt, hair (dog hair), etc. If it is dirty it will need to be cleaned, or just have someone clean the coil. You need someone that knows what they are doing so the do not ruin the coil or get shocked by the electricity. Do you know how old the unit is? To answer Harpers question Of May 19 -- yes high humidity and it's removal does greatly reduce the capacity of your A/C unit. The single large return in the basement is the "new normal" way to run returns in a house. I hate that installation because it is the CHEAP way to do a return. Returns for the top floor should be located at the ceiling and in every room, although this can't always be done. Another question I have is "how much insulation is in the walls if any and how much is in the ceiling". What type construction is your house. Measure the outside dimensions of the top floor, add a best-guess for the conditioned basement space. Trying to figure out what is wrong is hard for anyone on this site since they are not there. Lastly, use the psychrometer to check the basement humidity. I should use paragraphs in my writing but I was a service tech and not that good at writing. You may have gotten good advice from the A/C companies you have contacted or you may need to may still need to find that good one. Hope this helps

It's pulling air from outside

When the A/C starts, it's somehow pulling more humid air from outside into the basement. Hot air can hold a lot more humidity, so on a hot day, outside air that is below 50% relative humidity enters your cold basement and becomes air that is over 50% relative humidity.

Since the basement is cool already, decide whether you want the basement as part of your AC envelope, and if so, duct and seal accordingly. As you know, AC already dehumidifies, so running a dehumidifier alongside it makes no sense.