This is a battle you just aren't going to win with your current gear.. The bandaids mentioned above will help but it doesn't sound like that's a route you want to go.
I went down this road with la scalas, kt88 push pull amp, and Doge preamp. Tons of gain in the Doge. I tried everything and in the end it really comes down to the right equipment for the speakers.
Knowing what I know now I would have went low power with low gain or no gain preamp.
A solid SET amp with a high quality preamp that's low gain should get you there. Sayes or Radu amps/preamps have been fantastic in my systems.
Good luck on the journey, and congrats on the new little one!