Being a 12 year old scooter I think it would make sense to make sure the battery terminals (on the cables) are clean and the engine is properly earthed to the battery. A bad earth to the battery can cause such a problem.
The battery is new but are you sure the battery is fully charged ? Batteries hardly ever are fully charged when it comes off the shelve. Specially on a scooter I think a new battery should be charged before installation.
If it has not been charged properly and your scooter has a DC CDI, I think as a result of faster engine speed, more voltage will be drawn from the battery for feeding the CDI. Because the faster the engine speed -I think the more power “draw” from the battery.
Another possibility that I can think of - DC CDI’s have one or more capacitors inside the CDI’s. Capacitors never last for ever, and I think if your scooter has a DC CDI there is a possibility that a capacitor inside the CDI might be on it’s way out.
If your scooter has an AC CDI, this will obviously not applicable.
Not sure if this will help.