Well, lets see: a heat pump IS an A/C that can run in both 'directions'. You don't need extra electrical wiring for that (the heat) to work. It would be good to know if the unit requires both the neutral and the ground. It would also be nice to know how you connected it at the unit. Scenario 1: It requires two hots, neutral and ground. You connected two hots and neutral but no ground. It runs happily. But it is not a safe installation since the unit is not grounded. Scenario 2: It requires two hots, neutral and ground. You connected the two hots and ground but not the neutral. Then it would depend on whether the indoor unit uses a 120v blower. This probably would not work. but it might. Scenario 3: It requires two hots, neutral and ground. You connected the two hots and neutral. There is an internal jumper between neutral and ground. This is similar to how older 220v appliances used to be connected. It runs happily. IIRC, new code does not allow this type of installation (for appliances). The best long term solution will be to run the 14/3 from the box to the unit. Run it outside if necessary.

If your basement is mostly underground and you have one or two small windows then installing a mini split is over kill in my opinion. The basement requires no cooling in the summer. In the winter the heating load is much smaller than you think. This is due to the fact the basement is below grade, plus the installation you will be adding, and the first floor above is heated. How cold does the basement get in the winter, and how warm does it get in the summer? Adding some registers and a return is likely to be sufficient for most winter days. You could an add an electric baseboard heater as a supplement for cold days.

That’s my thinking, but I’m not sure how to handle it. The company that installed the system is reputable, and they seemed honest and respectful. But I paid a lot of money—over $19000 (although they called it ductless, they did install ducts in the attic and vents in all the rooms upstairs). When I researched costs, I based my decision on the higher cost of Carrier. The other estimates were higher; one was $30000! Even if Carrier and Bryant are the same, they should have explained that to me and indicated that they would be using Bryant. I plan to contact the company. I’m just not sure if I should ask for some sort of discount.

I agree with that ^^^ and appearantly you burned the toast. Odds of succesful diy electric work can be predicted by two questions. #1 Do you own or have access to a multimeter? #2 Do you understand what the various dial settings are for and how to use them? If there is a no in there anywhere,put the screwdriver back in your toolbox and call a pro before you cause damage that costs you 10x what a pro would have charged to begin with.