4" x 12" will give you a CFM of 175. 10" Round would have a CFM of 400. So using that rectangular vent would not work as you would have less than half the CFM's required.

Do it right, a 48" hood is usually quite powerful. My round 10" duct stays that shape and size all the way to the roof vent.

In the case of cooking ventilation, the most important aspect of hood duct diameter is keeping the air velocity in a range that minimizes grease condensation at the low end, and impingement collection at the high end. The recommended range is 1000 to 2000 ft/min. This range is higher than furnace ducting would be designed for.

Further, if the duct is cold or may remain cold with the cooking effluent passing through it, the recommendation is to stay closer to the 1000 ft/min limit. You may calculate the velocity by dividing the actual obtained CFM (see below) by the area of the round duct in square feet. Rectangular ducts tend to be barely better than a round duct equal to the smaller dimension, so a 4 x 12 is roughly equivalent to a 6-inch duct.

So you will have to stiffen up the ceiling joists to support a larger cutout.

But the cart is certainly in front of the horse here. Always start with requirements and then find the candidate hoods that meet them. Requirements are generally divisible into three categories: performance, aesthetics, cost affordability. I will deal with performance here.

You don't specify your range size, so I don't know if you are overlapping as needed for full capture, or not. However, as an example, assume a hood with a 48 x 24 entry aperture. This area is 4 x 2 = 8 square feet. Intake velocity should be around 90 ft/min; hence the total CFM when performing hot (oil or grease vaporization) cooking should be 90 x 8 = 720 CFM. Depending on make-up air (MUA) pressure loss, baffles pressure loss, duct pressure loss, and the actual blowers' fan curves, the rated combined blower CFM (at zero static pressure) should be roughly 1.5x this value or 1100 CFM. (If this is a Vent-a-Hood, we may need to address this more granularly.)

So, back to the question. If this performance requirement were met, then the actual flow would be 720 CFM, and in an 8-inch duct (0.35 sq. ft. sectional area) the velocity would be a bit over 2000 ft/min, so a 10-inch duct is recommended (1300 ft/min) as the hood manufacturer apparently indicates.

What is your MUA plan?