Are you going to leave this thick mixture in place for decades? Or is the idea to put it down for a year or two to kill weeds and then later pull it up and plant beneath? Best advice is probably to try an experiment on a portion of the soil.

As you noted, you're perhaps being a bit unreasonable - your two desires do cancel each other out somewhat; if you put a bunch of stuff on that's airtight, it will eventually go anaerobic.

But: my understanding is that the idea of smothering is mostly to kill the weeds by keeping out sunlight, to a much lesser degree air. Most of the materials you put above are not really airtight and will breathe - but to varying degrees. Very thick hard carboard won't, most fabrics will. Wooden slabs won't. Cardboard will get more permeable by air over time as it degrades - you can speed that up by poking holes in it (regular non-shiny carboard will probably become fairly permeable after a bit of time to break down).

A lot of soil life is very slow decomposition processes; air won't be needed in big fast quantities most of the time, and seeping through porous materials may be enough.

If you want a different suggestion, try putting some stuff under your fabric layers that is chunky (more air between them) and degrades slowly too. A layer of chunky wood chips followed by the other stuff would do. I'd go easier on the thick hard cardboard.

Just some opinions, good luck.

Thick layers of impervious materials cardboard could cause an air exchange stop. Landscape fabrics do not do that although plastic sheets will. Use of cardboard, newspaper, etc. to aid mulches in suppressing unwanted plant growth should follow reason, a layer of cardboard is of less detriment than would be an inch of the same thing.

am I also at risk, even a little, of subjecting my soil to those same
negative suffocation effects that are associated with landscape fabric

Some actual horticulturists say that you will cause decreases on oxygen levels, but there haven't been many studies.

A thick (4-6 inches) layer of wood chips will smother most weeds. So will a thick layer of gravel.

I did crude sheet mulching two summers ago to kill off area of lawn grass and turn it into vegetable garden. The cardboard was decomposed by the end of the season.

I’m not sure how recommended it is, but I’ve also used that red flooring underlayment paper that looks like large rolls of construction paper, laying it down on top of weeds and grass, and then putting a bark mulch on top. It did work, very few weeds came through, and it was decomposed within about two months. This was at the height of summer, in a fairly wet summer with a lot of rain fall.

The cardboard thing did a surprisngly good job of weed suppression for me. In my dry climate it was a couple of years before the weed suppression effect went away.