Given that it is a 16" lathe and you have the space along with 3 phase changes the picture a bit. You could easily get by with a 5Hp, but that 7.5Hp Baldor listed above is killer, but you do need an inverter/vector motor to achieve the speed range you want. You could use 2 gears all day long and get the range of speed you want. You need to match the VFD to the motor size, you cannot under size it by saying you are only going to use X% of Hp. Since you have 3 phase, just use a 7.5 Hp 3 phase input VFD and be done with it, you will also need a substantial braking resistor to go with it. Get a decent VFD with sensorless vector motor control, in my experience it holds better than 0.1% speed regulations under all loads, my lathe is +/-1 RPM. I have entertained (and had one on an install that I did similar to what you are planning) using a feedback encoder, but believe me it is truly wasted in this particular scenario and you will not see a performance difference. Feedback encoders are helpful where exact speed/shaft position is required, holding loads, very low speeds or absolute speed regulation, which is not the case. The VFD needs to be set for the motor operating parameters and autotuned to the motor.
You can then set the motor overload parameters to dictate how it operates and at what point it will trip (the default is usually something like 120% current for 1 minute at the particular operating output voltage). You need to keep in mind that below the base speed of 60 Hz, the motor will loose Hp in a linear fashion down to 0 RPM, so if you are running at say 20 Hz a 7.5Hp motor is delivering 2.5Hp. Above the base speed most of these inverter motors remains at full Hp up to 6000 RPM, although toque will fall off. This is one reason on factory installed VFDs that you often see an operating range of something like 20-180Hz. So most of the time you are operating at full Hp, low speed you have full torque. There are other factors, so above the motor base speed the applied spindle Hp is greater relative to a fixed speed motor and the torque is about the same as a fixed speed motor at high speeders. A fixed speed motor requires gear changes to achieve the higher speeds and the power/torque would be reduced by the ratio of the speed increase. The bottom line it will be way more than you can possible use, I have done quite a few installs with the Marathon BlackMax, Baldor IDNM, and a few other inverter/vector motors, they are truly amazing and almost not audible (most in this size range are TENV, so no fan). There is some tweaking of the VFD parameters to get the motor to operate in the sweet spot.
I would go with 3 belts of the current machine, should be more than enough to transmit the power needed and if anything I would rather have the belt slip/go as opposed to something else. You may be able to bush the current pulley, but decent motor pulleys are inexpensive and may be better balanced at speed. Pulleys also come in two piece with a locking inner hub that fits the motor shaft. Use a decent belt, I have been using Gates Tri-Power II that have a notched belt (they bend better and are less likely to take a set) with very good results. I really doubt you will have any shortage of power or speed range. I would stay with the current speed range and headstock oil weight, if splash lubrication you want to stay under 2000-2500 RPM. I see no reason to deviate from the manufactures specifications.