I was given my grandfather's WWII bringback P38 (AC43) a couple of years ago but I could not afford to even think about restoring it until now. Now things are looking a little up and I want to get it done.  It'll probably cost as much as a new gun, but...

It has some bad damage to the barrel and is never going to be a collector's item (i.e., restoration will bring back utility, not value) - but it's a piece of family heritage I would like restored to shooting condition and be able to it hand down as a functioning weapon to the next generation.

Can anyone recommend a reputable restorer?  I need to get a quote and put a budget together.

Look up Thor on this board, he posts in the Luger P-38 part. An exceptionally talented restorer of German pistols.

The gun rack in Florida http://www.thegunracksebring.com/  does barrel relining, even 9mm Luger I am told.  For function work contact Tom Heller (email is [email protected])    For cosmetic work, I can do the salt bluing required to restore it.  The other work should be done first.
Here is a code 480 Walther P.38 I did a few years ago as an example of my work.  BEFORE and AFTER pictures and a DURING restoration picture too.

AND a Zero Series Walther I restored sometime back.  BEFORE and  AFTER  Pics.   This one had some damage to the Walther Banner and heavily rusted in spots.

Do they reline rifles also?

I have a K98 that someone fired corrosive ammo out of and didn't bother to clean, the bore is rough.

How about a photo of the pistol so we know exactly what you are talking about???  
You have given no info except something about the bbl, what part of the bbl - inside or out???  
Depending on what the problem is you might be just as well off to buy another bbl (& put the orig away)???  
Sarge