I have no useful information, but this sounds like a really interesting idea. I hope you find some answers and share here how things turn out.

Have you considered installing a whole house fan? It would be installed in the ceiling below the attic. It would work better than what you described and easier to operate.

The hood is not designed to operate as a house cooling unit, it has a limited life.

Just install whole house fan if it works for you.

This is one case where I think separate systems for separate purposes is important. As mentioned above, get a whole house fan, you'll be much happier. Also, depending on the size of your house and climate you may want something more like 3,000 - 7,000 CFM or more.

@Charles Ross Homes, That is the kind of thinking and practice that's led to the U.S. having the worst indoor air quality of all developed countries and contributes to our highest rates of preventable chronic disease and lowest life expectancy.

We need more fresh outside air in our homes, not less. One of the biggest IAQ problems in the U.S. is that people keep their windows closed too often.

As to the FUD...

Whole House Fans only depressurize houses when windows are not open enough. Same for moisture in wall cavities.

Except for a tiny minority of cases of very high pollution, the air being brought in does not need to be filtered. In fact, it's much better for it to not be filtered!

I'm not sure I understand the intent of some of the comments. As I stated in the original post we blow hot air out of the upstairs windows while simultaneously opening all the downstairs windows to allow cool air to replace it. This method fills the house with cool air from the bottom up in the early morning when the outside air is coolest. I considered a whole house fan as we installed this in our last house and it worked quite well. Same principle - open all the windows downstairs while the fan blows the hot air out. I'm not sure how this got turned into an issue with not getting enough fresh air into our homes - we live in a mild climate and open windows to bring in fresh air all the time. This house has blown in insulation in the attic that is very light and is easily disturbed so I fear that an unducted whole-house fan would eventually cause the insulation to clog the attic vents. I suppose a ducted whole-house fan would work and I understand the reasoning behind separating functions. Since it seldom gets really hot we typically only do this for a small percentage of the days during the summer and it seemed to only have to run for about an hour so I thought it might be a good case for dual purpose. I just thought there might be some clever solution that could accomplish this without a lot of effort and expense. Thanks for the input so far.

OP, the fresh air comment was in response to Charles Ross Homes. The thinking he promoted that we should keep our houses sealed up tight and let the HVAC system take care of everything is exceptionally misguided, particularly with current ASHRAE standards and what we know about the affect of poor indoor air quality on occupants health.

Yes, you will likely need a ducted WHF. One option is: https://www.tamtech.com/product-category/whole-house-fans/

Trying to use the hood exhaust fan for double duty in this case comes with a number of risks and drawbacks that aren't worth it. One is that it is critical to keep the static pressure of the hood exhaust system as low as possible both for performance and to reduce fire risk. Introducing a damper setup would lower performance and increase fire risk. There is also the problem of a damper ceasing to function properly (which they actually do surprisingly often) and what problems that could cause, especially if you don't notice it immediately. It is likely possible to mitigate these and other concerns but if I were you I'd want to be cautious and enlist the expertise of an engineer in doing so. In the end you'd be spending a gob of money for a sub-optimal system. So good thought but IMO not practical.