First of all <HUG> because you need it! Welcome to the board. Of any where I have found, this is the board with the wisdom, knowledge, experience and heart to guide and support through the waters. Your story saddens me greatly yet uplifts me as well for the utter love that I can feel radiating out. I am impressed by the communication you have established.
Having not have been in your exact situation, I can only say that it seems to me that unless everyone is willing to bend and compromise a little towards one another that things may remain static. As much as Lily wants you happy and wants to express love as you do, there is something holding her back....maybe fear? Maybe that is something you both need to dwell into. She seems to be open to the whole poly thing and even wishes she could love like that which shows me that there is something more mental getting in the way. Losing you, being inadequate compared to Dahlia, seeign you two with something she's not involved with or part of etc. You don't relaly give a lot of info on Dahlia's mindset.
Have Lily and Dahlia spoken at all? They both know and accept one another but have they had a chance to relate to one another? Some would say it's not a good idea (I expect) but if I were one of those women, I would want to relate to the other person, to see what's there and to be able to make them a real factor. If they have never met or related, there is a chance that both have unreasonable fears of who the other is, what they possess that the other doesn't or a hundred other thoughts. We're all guilty of coming up with crazy scenarios about something going on that we know nothing about, it's human nature (IE: seeing cops outside your building and wondering what's happening).
New strategies are definitely called for if the goal is moving forward but be prepared that if Lily or Dahlia get to the point where they ultimately decide they can't or won't, then you will have to decide (with their input) the boundaries for moving on without part of the equation.
Good luck and hang in there!