This appears to because resolution seems to be based on the current class.

That in:
   final class IDo$$Lambda$1 extends java.lang.invoke.MagicLambdaImpl implements IDo

This call in val():
         4: invokespecial #19 // Method IDo.lambda$guts$0:()V

To this method in IDo:
  private void lambda$guts$0();
    descriptor: ()V
    flags: ACC_PRIVATE, ACC_SYNTHETIC

Fails with:
  Exception in thread "main" java.lang.NoSuchMethodError: IDo$$Lambda$1.lambda$guts$0()V
        at IDo$$Lambda$1.val(Unknown Source)

It appears to be trying to find the class in the current class rather than the class containing the private method.

A slightly altered version where IDo$$Lambda$1 does not implement IDo (and thus the bytecodes look identical) succeeds.

Attachments:
  Source files: Source_*.java
  javap output: *.javap
  Calling class: IDoLambda_*.javap (this is the generated output class for a lambda, but that doesn't matter in this case)
  Called interface: IDo_*.javap
  Code/output demonstrating the bug: *_Fail.*
  Code/output of the version that works: *_Succeed.*