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.