Nigel,
I've got a 3 hp 3 phase mill that's powered with a VFD. Some people seem to use that inverter term though. FWIW, You should see an almost unbelievable difference in smoothness of the motor drive, it will also be quieter, and your surfaces finishes should be much better. But, I don't agree with over driving any machine tooL much past it's factory design limits. Yes it can be done, but is it a good idea? Your bearings, spindle and pulley balance, and even the materials there built from were engineered on good quality lathes to operate at a specified maximum rpm. The higher you go with rpm then the steeper the cost curve gets as far as wear and parts replacement. They build and sell specialized very high rpm rated lathe chucks ranging from hardened steel into I'd imagine some fairly exotic light weight ultra strong aircraft grade materials for very high rpm work. But those specially balanced high rpm steel chucks alone have prices to match. Yes I know you said you were planning on using collets, but I'm trying to show how expensive something like this can become, and how quickly it can start to add up.
Unless you do a great deal of very small work the costs to do this may not be worth it. And an inverter or VFD or any currently available pure electrical method will not give you full motor torque as you lower the rpms. That still requires it to be done mechanically. Whatever you do there's always a trade off that has to be done. IMO, the inverter/VFD is well worth it for the variable speeds, smoother running, better finishes, and maybe a very limited use of the higher speeds once in awhile. But if all your wanting is the high speed for small diameter work, then I think I'd be looking at something like a spindle speeder that will give you a way through gearing to increase the part rpm without increasing your main spindle speeds. And if you really want high speeds, they make very specialized air driven turbines that will do well beyond 50,000 rpm.
I have to think that there's not a quality lathe manufacturer anywhere in the last 50 years that would say it's a good idea to increase the spindle rpm by 25% or more over what they designed there product to do. 4,000 RPM today is actually very slow if you look at what a lot of the newer CNC equipment is capable of. The difference is it's designed from the start to do so.
And I've yet to operate any Super 7, but does yours have a proper roller bearings at both ends of the spindle, or is the front bearing the usual drip fed oil lubed solid type? Burning up a bearing and/or spindle journal could get a bit costly. I'm certainly not trying to say your idea is wrong, but there's a lot more to this than just replacing the motor on your lathe.
Pete
.