Can I convert a split phase motor to a permanent split phase one?
A split-phase motor that does not have a capacitor, is designed so that the starting winding has a higher resistance to inductance ratio than the main winding. That is usually accomplished by using a smaller wire size for the starting winding. With that design, it is essential to disconnect the starting winding after a few seconds to avoid overheating the winding. If a sufficiently small value capacitor is inserted in series with the starting winding, the overall impedance of the winding could conceivably be raised to the point that the current would be sufficiently low to permit operation as a permanent split capacitor motor. However, that small current in the starting winding would likely not be sufficient to start the motor.  The normal current in the starting winding during starting would be several times the rated full-load current of the motor. A safe current in the starting winding for continuous operation would be a fraction of the rated full-load current.

The easiest way to use the motor without the starting relay would be to use a momentary-contact start switch that would connect the start winding only as long as you hold the switch closed.  I have operated a machine that had a manual starting arrangement like that.

Re comment question:

The momentary-contact start switch would connect the start winding in the same way as the starting relay. It would connect the winding directly to the power supply without a capacitor in series. Both windings draw high current during starting and the current in both windings is reduced as the motor approaches full speed. However, the start winding is not large enough to continuously carry even the reduced current that it would draw at full speed. Here is a diagram: