Going to agree with everyone else, the 22" for ducting seems CRAZY! We run both Heating and Cooling systems here, and everything stays within the conditioned space. I understand your struggles though. When I brought my plan to local HVAC companies, I was quoted $12k for the system and EXTENSIVE bulkheads throughout the house. It was CRAZY. Thankfully my father works at a college that trains HVAC specialists, and when he brought my plans to the instructors there, they created an AWESOME system, with ZERO bulkheads, utilized all the interior space (just a small double wall in the middle of the house for the main trunk to come up) and the final system was just $3000 including the furnace and all ductwork when we installed it. Tested all the loads, temperatures and velocity of the air through the system in all outlet vents last night and the guys at the college ran the performance numbers this morning. Just need to turn down one damper and I have PERFECT heat and circulation in the entire house. Sometimes I just think it's an easy way for these guys to make $$ and get a bigger job by suggesting these crazy (but way easier) systems. It certainly would have been easier to assemble and design with a giant void in the house or bulkheads everywhere, but I'm not paying them 4x the price to make life easy on them.

I think other are saying to possibly use rectangular "inside" the wall behind this cabinet that is between your kitchen and DR. If that is the case, I do not think that will work or is necessary. 1st you would have to bend/transition to get that into the wall cavity. You want to eliminate bends as much as possible if you can. Not to mention you will have to take a nice chunk out of the back of the cabinet to accomplish this I would think. Not to mention I think you need something like a 5x12 rectangular duct (narrowest) for the same cubic foot of air movement for an 8" round (according to charts I have seen on the web). This will not fit in the wall. If I am looking at the 2nd photo in your attic correctly, you are saying on the right side on the photo, above the electrical junction box, that is mounted on the joist, is where the hole is located. There is another joist behind it that is the attachment point for the DR ceiling Drywall. There is block on top of the joists connecting the two and there also appears to be horizontal blocking between those 2 joists as well. So again not sure anything could be run inside the wall. If the walls are 2x6 then you have a shot. You said you have 6" from the edge of the joist to the front of the cabinet. So you may be able to transition something here within this space. I brought up Oval duct since it has much better/smoother flow, compared to Rectangular. Also, it would be 3 1/2" width and you should have about 3" now, from the front edge of the circle to where the wire was hanging down. Thus, you would only need another 1/2" of cabinet cut to fit the duct towards the front. But the duct would require you cutting wider.

" Just curious, if a "full basement foundation" is built on grade, how is it a foundation rather than a first floor/ground floor?" If you can dream it, it can done. My intention was to artificially raise grade, by grading up to the 13' higher second floor, making the second floor appear as a first floor, which would no doubt require thousands of yard of dirt but I had access to free dirt; the size of the lot made it feasible. Visually it would appear as a raised home or on a hill, with a driveway leading up; not an uncommon sight when I lived in NJ. I was keen on a full basement to accommodate a couple of bowling lanes, something I may vaguely consider building as a separate inconspicuous structure somewhere on the lot, however there are other important things at hand now, like completing interior design and renderings; these are close to getting finished, just need to complete designing builtins around the fireplace then the walkin pantry to DW's specific requirements.

The next question is, where would it have gone otherwise? If the builder ignored several obvious shorter routes for the crawlspace, maybe it is a questionable decision. But if it goes through the crawlspace to avoid blasting someone in the face on a deck or patio or to avoid a vaulted ceiling upstairs from the kitchen or some other convoluted path, maybe it is the better of several not great options? (My house has corner to corner glass on both front and back elevations on the first floor and a glass wall with exposed sheathing ceiling on the third floor. (And common wall houses on each side). There are very few places to run any ducts.We are fishing a vent duct straight up and through a knee wall that avoids any of the glass walls, but my neighbors have one duct that bumps out in front of one of these windows, and another vent that exits exhaust right next to their front door. Not attractive. We are able to run this duct relatively straight because we are demoing back to the masonry wall between houses.)