I have said it before, but I will repeat myself again.
It is not about weight, it is about mass distribution. You have the same problem with most commercial swords, but it is much worse when you are talking replica rapiers, whether they be blunt or sharp. I have held extremly light historic rapiers and extremely heavy historic rapiers, but they handled well despite the weight because of their mass distribution.
Most commercial rapier blades I have seen or handled start with 1/4 inch flat stock and either forge or grind them out depending on the factory/smith. They don't have the correct distal taper or enough beginning mass to handle correctly.
If you want a better blade that would be fairly accurate historically, start with 1/2 flat stock, grind a distal taper into it that goes about 2/3rds of the way towards the tip where it will reach a thickness of somewhere between 1/4 to 3/8ths of an inch, then level off and run at that current thickness towards the tip. At the very tip, the blade may even thicken some to support a stronger tip that will not easily bend over if it hits a hard object during a thrust.
You will also find this type of distal taper in historic longswords, arming swords, etc though the blades are not ususally as thick at the cross as a rapier, often being somewhere between 1/4 inch to 3/8 inch.
This type of construction will go a very long ways towards giving you the proper mass distribution for your replica rapier. However, the cost of grinding away that much steel, or forging out that strong of a taper takes a great deal more time and effort, therefore costs would go way up.
all the best.
Brian Hunt
P.S. I have handled many extremely floppy sharp replica rapiers. Some of them have had even more flex that some schlager blades I have handled.

