I am an experienced carpenter, and have built 3 houses, turn-key, by myself. So cutting into a joist is no big deal for me. I can see where it might intimidate someone not so experienced.
The KMart fan sound like a jet engine? No it doesn't, it isn't very noisy. Every night in the summer I run one or two of 'em in my bedroom. Bedroom is on the second floor, I put them in a window blowing out, and open 3 windows downstairs. Cool air is sucked into the house all night, and hot air is blown out the windows upstairs. Cool the house down to 68 degrees, shut the house up in the morning, don't need A/C. This is in the cool NC mountains.
I can run two of these fans in my bedroom, and you say they will sound like a jet engine in your basement? That is ridiculous.
As for, "You get what you pay for" well, maybe. I helped my neighbor build a big 1,900 sq foot log house. Tom liked complicated stuff. To cool his house down in the summer, he bought two fans on line for $500. We put a "box" in the top of the gable end of his house, where the cathedral ceiling was. These two fans were thermostat controlled, about 18 feet off the floor of the living room with the big cathedral ceiling.
The box we build was 30 inches long and 15 inches high, the fans were about 15 inch diameter. To keep bugs from coming in in the daytime, there were louvers on the outside that closed automatically when the fans shut off.
Tom spent about $300 in additional materials and labor to get these fans installed, total about $800.
The fans worked well for about a year, and then they conked out. The company that sold the fans was out of business.
Before Tom and I began working on this fan install, I told Tom how I put a $15 Kmart fan in the bedroom window at night, worked great. Tom was not interested in a Kmart window fan.
Tom got what he paid for, a couple of $800 fans that didn't work. Today, Tom has a 15 buck Kmart fan in the window of his upstairs bedroom. And, in the winter, he loses all kinds of heat, through those metal louvers on the doomed cathedral ceiling fan project.
Never forget the Great Axiom of Engineering, "Simple is better than complicated."
Also, you say the fan will blow cool air? That assertion is ridiculous. The fan will be mounted in the ceiling of your basement, where the wood stove is. The basement ceiling is where the hottest air in the house is, the fan must, and will, blow very warm air.
Good luck with your project, sorry I can't help you, looks like you and I are on different wavelengths.