Foam by itself probably won't do much to help. The Bianchi frames you refer to are using much thinner wall tubes. The foam used there eliminated a "ringing" of the tube itself. (The tube walls would literally be deforming, so the foam damped that motion.) Clip-ons don't ring, per se, but rather transmit, linearly, the vibration of the bike. Bar ends help because the energy of the vibration has to accelerate more mass, so the amplitude of the waves is smaller.
On the other hand, if you had some mass embedded in the foam, the energy of the vibration would be absorbed by trying to move that mass within the foam. So, if you took some Great Stuff expanding foam, and could somehow get lead shot into it while it was setting up, that would probably work very well. Expanded polyurethane like that has pretty good damping properties. You just need something for it to damp.
This is a great explanation.
But let me add just a little.....
Basically every substance in the world has whats called a resonant frequency, or basically a frequency that makes it vibrate. The more dense the substance the harder it is to make it vibrate and the lower the resonant frequency.
With this in mind think of a bell on the counter of your local dry cleaners. You ring the bell and it vibrates at a high pitch but if you lightly touch it with the tip of your finger the tone changes and the frequency lowers and it won't ring as long. This is a result of making the bell more dense.
The same thing happens when you ride your bike. The vibration you feel is the frequency the bike is transmitting at a given RPM and you can feel it in your hands when riding. I'm sure it feels different as the RPM's change. If you change the resonant frequency of the bars by changing it's density then it will not vibrate at the same number of cycles per second and will feel different on your hands. A higher frequency will feel annoying and buzzy and a lower frequency will go mostley un-noticed.
The idea of the "bar snake" is to lower the resonant frequency by making it more dense and giving the illusion that the vibration no longer exists.
So the short answer is adding any substance to the inside of the bars will most likely change it's properties and give you a different feel to the bars as you ride.
What I did in the past that worked really well was to use a silicone caulking and lead shot you can buy at any Dick's sporting goods in the hunting dept.
Take the lead shot and mix it with the caulking and then insert it into the hollow bar. Put some tape on one end to stop it from leaking out or use the barend weight wrapped in something and lightly tightened to allow it to be still used inside the end of the bar. Let it dry in accordance to the instructions on the tube of caulking and then remove the barend and take off whatever you used to wrap it and then re-install the bar end and you're all set to go!
