Run 1800 RPM motor at 3600 RPM
I believe that NEMA standards require some mechanical overspeed capacity. I may be able to look that up later, but I am guessing it might be 25%. Based on various projects that I have worked on or quoted, I think there is a very good possibility that 50% overspeed will be ok.

The primary concern will be rotor balance and the balance of the external fan. You might consider removing the external fan and replacing it with a separately powered blower. There is a good chance that the bearings will be ok, but you need to be concerned about overhung load.

You probably already know that the torque will drop drastically unless you can increase the voltage as the speed increases. You may be able to get constant power up to 150% speed, but the available torque will drop at a faster rate above that.

NEMA Standard MG1

The following excerpt from the condensed version of NEMA standard MG1 indicates that the motor should be fine at 3600 RPM. Note that the table is for direct coupled applications, not driving the load through a belt.