Deb
Do not use any more drain cIeaner. It does not work. It is caustic. It can damage pipes, snaking equipment, and people.
I would pull the drain plug first to check for hair and stuff (it is amazing what you find down drains), but I wouldn't try to run a snake through the lav tailpiece and p-trap--really tough to get through those small fixture p-traps and a lav could have 1-1/4". To remove the stopper assembly, you have to remove the rod that is holding the stopper down. If you have hair and soap accumulation here, you can see and it may be necessary to remove the p-trap to clean anyway. If this new construction, you could have some kind of construction debris. And, like you mentioned, it could be an installation problem. Go to the next paragraph.
If that is not it, I would next check that the horizontal piece of the p-trap that goes into the wall is not too long. You often have a 90 in the wall. The OD of the p-trap is such that is will go inside 1-1/2" drainage piping. The horizontal part of the p-trap can be inserted into the pipe in the wall so far that it is against the back of the 90, cutting off almost of the drainage area. I
have run into this more than once.
Deb
The Pipewench