I'm not following...
The copper tube should be nominal OD dimension with a very small tolerance. In smaller sizes they are common in 1/16 inch sizes, so your open end wrench set makes a great size gauge for the tubing.
When you put the two together, you have slop where? What piece is 0.044 larger than 3/8? Tube? Ferrule? Fitting?
3/8 inch is 0.375 inch. Add .044 to that and you have 0.419, which is awful close to 27/64 of an inch, which is a really odd copper line or fitting size. Unless you meant 0.44, which is might be 7/16 (0.4375), which is a lot more likely to find. Or you mean your existing line is .044 smaller than a 3/8 fitting? 21/64? Again really odd to find. I wouldn't even consider switching to plastic line until you were sure just what size line you were replacing... Once you know what size, I can't see any reason not to go back with copper. The line and the fittings at either end of the line are all identified by the nominal copper tube outer diameter.