Drop USB voltage to 4.5v
However I'd like a solution that can sustain 4.5V regardless of the
  current (within the limits above)


If you are happy with a constant voltage of 4.5V then you should consider a low drop out voltage regulator like this: -



It can be setup to deliver up to 4.5V and at 1A to the load and will only need 4.645V (typically) on the input to sustain the output voltage. Note that it needs a minimum load for it work correctly (1mA) but this shouldn't be a problem given your requirements.

This is typical of a series of many voltage regulators with low drop-out voltage - I'm not saying use this one - I am saying be aware of what TI and other folk may be supplying. It's likely you can find one with current limit circuits built-in. You could also apply a bit of current limiting in this device by having a resistor in series with the voltage feed - this should be chosen to develop just enough voltage across it (at the required load current) to sustain the output. Should load current increase, output voltage will drop.