A quick update:
I think I have now been able to fix the problem: in the case of a member access, the CompileVariableAccess method does not work (which is not surprising I guess. It seems this one is defigned for variable access inside of a function right, not accessed as a member, except maybe if written this.myMember, right?). So I have gathered the code managing member access (the one managing 'ttDot' in CompileExpressionPostOp) into a method that I call instead of CompileVariableAccess, and it seems to work.
Still needs some more testing, but if it runs fine I'll share the new code. Given my limited understanding of the compiler, there is probably a better way to do it though, but I might need some help to clean it up!