Ok. I heat from the basement too, but to keep my main floor around 70 F, the basement temperature is much higher. I'll explain my "air moving" strategy below, but the result is that it's 85 F near the basement ceiling and 77 or so at the basement floor.
If you want to be comfortable in the basement AND heat a large home above the basement, you'll have to compromise some. It won't be possible to keep both at 70 F or so.
Now, I have central stairs to the basement, with on the main floor on one side of the stairs a living/dining room and an open kitchen. On the other side a hallway, bathroom and 3 bedrooms. (Then two more bedrooms and a bathroom one floor up.)
I ran with the heat coming up through the stairs one year. It was not possible to heat with wood alone this way. I think it's because what happens is warm air rises through the stairway, and cold air (necessarily) has to go down through the same stairs. So eddies appear, and total heat flow to upstairs decreases.
So, I made the following. Before do anything like this, talk to your code inspector or fire department.
I made a register on the far end of the living room floor. To that register I attached metal ducting and a fire damper (device that automatically closes if it gets too hot), then a flexible duct down along the wall of my basement and an inline fan (and register) at the floor of the basement. The fan sucks the (coldest) air from the floor of the living room and deposits it on the floor of the basement where the stove is. This pushes up the warmest air near the ceiling that then rises to the main floor through the stairs.
The important parts are the metal ducting and fire damper: you're making a hole in a floor. This helps air, but could help fire spread much more quickly too. Even wooden floors have a fire resistance (i.e. a time in which they delay the spreading of the fire - this delay due to the fact that they have to be burned before the fire can get through). Hence code, FD.
The second part is the fan: have it run the right way, depositing cold air on the basement floor. This keeps the warm air near the top, keeps temperature stratification and heat rising to the main floor.
The third part is the flexible duct: you don't want any fan vibrations transmitted to the floor (joists) of your living room floor.
This set up allows me to move enough heat upstairs. But it makes the basement still very warm.
So if you want to spend time down there AND want the main floor to be comfortable based on wood heat, you are having a problem. The problem is that a stove or insert is a space heater, and thus one often (but not always) runs into trouble if one wants to heat more than the space. In particular for a home that large when a LARGE heat flow has to go up the stairs, needing either a huge air flow (large fan noise), or very warm air going up.
Me thinks you have to compromise: get the insert to get the basement to normal comfortable temperatures, benefit from any heat that naturally rises to the main floor, but don't expect the main floor to be at temps from the insert alone.
9x13 would properly allow for an insulated liner to be installed. That's good.
Finally, I don't think you have to choose between "heat source" and "fire view". The BK has flames too when running above midway thermostat settings. And other stoves will put out the same heat as a BK (or better).
The BK is "different" in that it has extended the low end of the BTU output range.
I don't see the contrast you draw between "heat source" and "seeing fire". For most stoves (including the BK in its upper half of the range), the heat is coming from the fire. So they go hand in hand.
But first you have to decide what volume you want to heat, keeping the above basement/main floor temps in mind. And then determine the BTU capabilities you need and then look at what pleases your eye and fits.