In this case, Twowaxhack, we are talking mm of water pressure, not inches of water pressure for this installation. And as your design has the same issue with compressing the air in the "vent system", I'm not sure why you don't understand that the more air in the "vent loop" you have, the less pressure that will be exerted on the water in the trap. Nor do you appear to understand that the larger surface area you have at the top "vent" connection using a 4"x4"x4" wye, the better access the "vent" has to the air in that vertical run. And this design would not require an AAV, although you could put one in at the top of the "loop vent" if you really thought it would be a benefit.
This is the difference between "plumbing theory" and actual "engineering principles".
Bomaifam, Based on Twowaxhack's "Goodbye, forever" comment, he may have blocked me. So do as you like and use whatever design you deem has more merit. He is a Professional Plumber, and I am a Professional Engineer. This is not a code issue as neither of these designs meet the code. One last thing that needs to be checked. As this building has an upper and lower unit, my assumption is that the basement is common. And from your pictures, I'm assuming that both units share the same sewer outlet where you are wanting to install this sink. If that is the case, when you start into this project, be sure you coordinate with your upstairs neighbors and that you have everything you need to complete the job in a short amount of time.
One suggestion would be to pre-fabricate an assembly including a new cleanout with the two 2" stubs capped off. Something like the sketch below. Then all you need to do is to cut the existing stack in two places and install the assembly with a couple of shielded rubber couplings. The sink and all associated piping up to the capped stubs could then be installed at your leisure, instead of in an urgent time frame for the entire job. Then at a second short "shutdown" of the sewer system will allow you to make those two final connections.
View attachment 32657
Good luck and God bless.