There are several things you can do to increase the lift of a helicopter.
1. Increase the RPM of the motor. To increase this would require you to increase the power since V x I = Power raising either the current or the voltage could accomplish this (however the DC motor may or may not be able to take an increase in either). If it cannot take this increase, it may be necessary to use a larger or at least higher power motor.
2. Decrease the weight of the helicopter. It appears you’ve already tried a bit of this so it may not be enough.
3. Increase the lift provided by the blades. I’m not sure what model of helicopter you have so I don’t know whether or not yours has variable cyclic or not. You could change the curvature of the blades or (if variable pitch) increase the pitch of the blades.
All of these methods would require some toying with the actual mechanics of the helicopter. It may just be easier to return it for a different one.
As a side note: You said 6 inches long. That is a very small RC helicopter. Is it by any chance one of those quick charge capacitor powered mini RC’s?