Warning: Thought experiment ahead, not intended for real battles.
I started looking at the weight difference between foam weapons and historical weapons. Putting aside the rather extremely late medieval weapons, it looks like 2-3 pounds for a medium-length sword is realistic (http://www.hroarr.com/longswords-and-their-data/ say down to .94 g/mm == 084 oz/in), while ours are in the area of 1/2 - 1 pound (1/3 oz/in). "Realists" complain that real swords could not be used the way we do, but actual steel swords, even blunt, could not be safely used in our type of battle.
So could we make a sword that's safe for our style but heavy like at least some historical weapons of the same length? Here's some thoughts on construction:
Using a plain metal core would give 1.3# for 1 meter of 10mm round rod. The distribution would be pretty awful compared to real swords, though, and making the metal tip safe would be difficult. The heavy tip would also pack quite a hit.
Maybe using a fiberglass core but glued inside increasing diameter metal tubes towards the base. This would add some weight to the overall sword without making the tip heavy or dangerous, and might simulate real sword weight distribution well. The end of the innermost metal tube would be the weakest point.
To make a more durable construction, and to make the tapered core easier to deal with, maybe the innermost part of foam could be molded latex or even calimacil. Did anyone experiment with latex on the inside?
I may have to make a pair of these, to test in 1:1 battles. I still hold that actual warriors of the time were simply stronger and could handle heavier swords more easily than us desk jockeys, but it would be fun to try.