Loss of mid-tones when using lith film is most likely due to
developer choice. A huge number of people use diluted paper
developer, why escapes me. The need is for a softer working
developer. Since paper developers are many times more active
than are film developers, a diluted film developer makes far
more sense. If you don't believe this, try developing paper in
full strength film developer. Over the years I have used HC110
1+127 from syrup, D-76 diluted 1+4 or 5, D-23, among others.
Currently I use Somarkos LC-2 which is superb for this process
providing tremendous control over both the positive and the new
negative.