Compatible Coaxial Fittings for AT&T U-Verse Cabling inside my House

I assume I can use off the shelf coaxial cabling for this purpose,

For another tv to have uverse access you are going to need another att receiver ($10/month).  Unless you have an HPNA setup (uses coax) which these days is the exception, not the rule, you connect att receivers with ethernet (from gateway to uverse receiver) if you want a wired connection.  Alternatively you can get a att wireless receiver and don't need any cables to that other receiver at all.

I seem to remember when AT&T did the initial install that they had to replace all the existing coax connector fittings to allow for two way flow of data?

Ok, so I guess you do have HPNA.  But att didn't change any coax fittings.  Coax is coax with standard coax (F) connectors.  If all you want to do is change where the existing coax runs to the current gateway location then run it to the new location instead.  If it isn't long enough to reach the new location then use a female to female F-connector (example) and a coax cable of the appropriate length to reach the new location connected to the end of the exiting cable with the coupler.

I may not fully understand your problem as to why you are thinking "y-cables" or "t couplers".  I think that's why I initially thought you were trying to hook up a second tv.  Guess not.  It's one coax going to one device (gateway in this case).  No reason to do anything other than extend it to the new gateway location and connect your dvr to the gateway in the same way it is connected now.

The they preferred to replace your existing F connectors with their F connectors just to make sure the connectors were, well, connected properly.  But they are still garden variety F connectors.