Walaikum salam,
MashaAllah congratulations for the car. Yes, you can go to civic center and get the car registered your self without any middle man or bribery. The last time I went to register our new car was in 2020, during peak Covid times. The public dealing timings were limited. However, the work did not stop. Don't know what their workings hours are these days.
Not sure about the charges of a 660cc, but if a dealer or an agent has quoted you 6K then the actual cost of registration would be around 3K. For eg; I had to get a commercial changan pickup truck registered as a filer. The dealership quoted 34K for the registration and commercial permit, I declined and went there myself and got the car registered for approx 23K. Around 11K difference.
When you enter the building you will see a few photocopy shops towards the left of UBL bank branch inside. Ask them you want the form for vehicle transfer. The form is only a few rupees and has all the requirements mentioned on it. People there also guide on every step. Will probably take your 3 to 4 hours of the day but it is doable. Also, the vehicle inspection is just a cursory look at the vehicle by an inspector in the parking ground next to the building. Sometimes, they don't even do physical inspection and just clear the vehicle. Just keep all the vehicle documents, sale deed, registration book, buyer and seller nic copies with you.