There is no way that a simple fireplace insert can heat a house the same way that a central air system that pipes the heated air into each room.
It is possible to have a fireplace built that has a heat exchanger built into it that feeds into an air circulation system such that the warmed air is moved around to other parts of the house. However I doubt that your existing fireplace has this heat exchanger system built into it...then yet the ducting to an air curculation system into the rest of the house.
I'm glad to see that you seem to have it in mind that the advice that you get from the fireplace salesman will always be biased toward what he is trying to sell you!!
It's a balancing problem, fairly typical for a one-zone heating system spanning an entire house.
Mind you, I'm a bit concerned with a setup that kicks on the A/C in heating season - being from a primarily heating climate, we typically have a "winter/summer" or "heating/off/cooling" mode switch either on the system/boiler/furnace or on the thermostat to prevent that sort of foolishness. If it's REALLY so hot in winter or cold in in summer that we want to change the mode, WE choose to change the mode, rather than leaving it up to a machine to decide.
Presumably you have forced air heat, since the same system is cooling. One approach to improve balance is to run the fan more to distribute air around the house and balance temperatures even when heat is not being delivered. A more basic step is adjusting the airflow to different parts of the system for heat delivery that more closely keeps things even - but if the same system is cooling this may be difficult to get balanced in a manner that works well for both, since heat rises and cold air sinks, left to itself.
The system may be oversized (so it quickly heats or cools the location of the thermostat, and then shuts off the fan, rather than running a large percentage of the time when it's cold out), but all systems are prone to being somewhat oversized much of the year in order to be large enough to heat on the coldest days and cool on the warmest days.
A good HVAC professional may well be worth talking to in order to tune your system as best it can be tuned. Moving to continuous circulation (perhaps at a lower fan speed) seems like it might be needed in this house to reduce stratification, at a guess. The only major downside if your system is not too noisy will be the electric bill for running the fan.
As a quick stopgap, examine your thermostat to see if it has a "fan" switch, typically with two positions - Auto (blows only to heat/cool) and "on" or "Continuous" - if the switch exists and is wired correctly, that should put the fan in continuous circulation mode - but you may want to alter the system to make that quieter and/or more efficient with a lower speed or even an entirely different fan/blower. And it may still need to be balanced to work better.
Best Answer
Have you considered putting in a wood furnace instead of a stove? You can get models that incorporate a boiler, so with some clever plumbing you could use wood to heat the water going to the baseboard radiators.
And if you're not opposed to running some duct work, you could run the heat from the furnace to both floors.