Wireless / Wired access point

I have a friend who I set up a video security system for in his small business.  His cable internet comes in on the top floor and that is where is router is installed.  He uses it for about 4 devices on the top floor, some wired and some wireless.

He wants to install a computer and a wireless Point of Sales terminal on the ground floor shop.  The wireless signal is sketchy by the time you get to the ground floor.  I was thinking of installing an access point for him.  I am unable to find the specs I am looking for, but think that maybe they are just advertising the products the wrong way.

So here is what I need.

A Wireless access point that offers wifi extender type services.  Meaning that is should extend the same network not create a new one (like connecting a wireless router would do).  I can not have double netting, it will break the security system.

I need it to have at least two wired switch ports that again, will only be switching, not routing (NO DOUBLE NAT).

I would like to spend as little as possible but get something decent.  Maybe the $50 - $100 range.

Any ideas?