Recirculating vent systems exist. To be effective they need as much flow rate as a normal vent system for the hood size used, and the air must be passed through a series of filters before being returned to the room. Ventahood ARS is one such system.

https://www.houzz.com/discussions/2347276/vent-a-hood-ductless-ars-range-hood-update#n=65

Is there no way to run ducts to the roof or along the ceiling to some other exterior wall? Sometimes, you can even run ducts in the floor, but there are very real issues with that, so you'd have to consult an experienced HVAC contractor who understands about vent hoods (not just about forced air heating). The main concern is about grease build-up, as residential ducting usually doesn't have serviceable grease traps.

In general, while short runs are ideal, longer runs are still better than no exterior venting. And it sounds as if you are OK with making bigger and more invasive changes to you your home. So, I'd try to think a little bit outside the box to find a way to route the duct.

I have a hard time picturing how you can have a unit that doesn't have any exterior walls nor a roof. But I can believe that your kitchen would be in the center of your unit. And I can also see scenarios where running ducts through other rooms is simply not doable. So, if you tell us that you ruled out all these other options, I'll absolutely believe you. And in that case, @kaseki's advice is very good.

Recirculation with either charcoal filters* or plasma filter.

(* need to replaced at least once a year or when smelly)

Make sure to pick the hood suitable for your kitchen volume (or bigger when open plan).

I agree that the charcoal filter ones work a bit better but you need to make sure to clean.The vent-a-hood one is supposed to be very good.

Yes, the kitchen is in the center of the unit and has no outside windows (only enlarged pass through wall openings into the next rooms). One side of the kitchen there are 2 rooms before the exterior wall and the other there is one room before hitting an exterior wall.

To get outside on the main level, best case/easiest scenario we'd have to do at least a 15' horizontal run through the common wall we share with a neighbor before we'd hit an outside wall of the building. Going up through the roof, we'd have to once again go through that common wall up to the second story and come out through the roof somewhere. Our having touched the roof would then make us responsible/liable by the association if there were any current or future roof problems. AND, in either of those venting scenarios we'd first have to obtain association approval for this structural work, something we have no idea how difficult it might be.

Can you see why I am thinking that non-vented is going to have to be our route?

Thanks for your ideas.

Have a OTR MW "vent" as well-- considering remodel down the road so following for info!