You e got a bigger problem. No wood should be visible in that pic. It should all be stainless. Your liner is not big enough. Who is designing and specifying this? Because if your range is powerful enough even consumer grade, youre going to need that more powerful hood, and the MUA that goes with it.

It looks like replacing cabinets would be a last resort. Vent-A-Hood has a 9" tall hood that can be ordered with a 300 CFM blower. The hood is only 21" deep front to back so if you need it 24" you could pull the hood out to the depth you need and have a metal filler made to fill the space between the hood and the back wall. The hood is the CWEH9-142 here is the link to their web site. https://ventahood.com/index.php/support/specifications?id=361&view=article&model=CWEH9-142

Why not cabinets 18” and 30” or 12” and 36” to get you to 32” above the range? Can you reach the controls at that height?

No air leaves the kitchen that isn't replaced by some form of MUA, whether deliberate or house leakage. While code may draw a threshold at 400 CFM (rated), the actual CFM achieved without deliberate MUA is likely to be down in the 200 CFM region.

For capture of cooking plumes driven by hot oil cooking (stir frying and searing), with plume velocities potentially over a meter per second, the specific flow rate should be upwards of 90 CFM per sq. ft. of hood entry aperture (the entire area) for common residential hoods. And this is actual CFM, provided by a blower that has been selected after accounting for pressure losses in the cooktop to outdoors back to cooktop loop. These include baffle, duct, and MUA pressure losses vs. the blower's "fan curve." See sketch.

If you don't have any other combustion appliances, or have them but they have their own MUA and are thus not affected by kitchen air pressure, you may be able to size "passive" MUA (no blower) vs. candidate hood system blowers such that you can achieve the desired flow rate calculated from multiplying the hood entry aperture by 90 CFM/sq. ft. At a minimum passive MUA will have a filter pressure loss, a possible heater filter loss, some (maybe small) duct and diffuser pressure loss.

With affected combustion appliances and without means to provide them MUA, then MUA for the hood system will require balanced pressure in the house and this will require that the MUA system have a blower.

In general, determine requirements first and choose components second.

Thanks for everyone's input. The builder is going to give me a quote for installing the MUA system and changing the vent to one that meets the requirements of my gas range. I think it's probably what needs to be done to resolve the issue.