convert capacitor motor to three-phase
Whether or not it is worth the effort mostly depends on how much education or enjoyment you gain from the project. It is certainly not something that will provide something useful for less than you would have to pay to buy it as a new motor and variable frequency drive (VFD or inverter).

If the motor has a compatible number of slots, it can certainly be rewound for three-phase power. Winding it for 12 volts may be quite a bit more difficult than 60 volts or 30 volts. At 12 volts, the wire will be quite a bit larger and one turn difference in the number of wires in a slot will will be more significant.

You probably can not get very far with a project like this without good reference material.

Most questions about how to do it will be way too broad for this forum.

Re Comments:

Winding the motor for three-phase should provide a motor with performance comparable to a motor designed to be a three-phase motor. Winding for a lower voltage will be similar, but the internal voltage drops in the motor will increase significance as the voltage is reduced. VFD operation always faces that problem. At higher frequencies, constant V/Hz provides good performance, but but the V/Hz needs to be increased to compensate for internal voltage drops at low frequencies.

The DC power supply can be very simple. All that is needed is a transformer, rectifier and capacitor plus a system to limit the capacitor charging current.