The 3 1/2″ Grayson lathe can use gears from the Myford ML7 / S7, which are readily available. Myfords use a keyway system. For the Grayson, you will need to drill a 1/16″ dia. hole to suit the Grayson dowel pin system. Ignore the keyway.
By all means acquire a 3-jaw chuck, very convenient but you may have trouble sourcing a suitable backplate. The thread is similar to a Myford but the register diameter is smaller, 1 1/8″ vs. 1 1/4″ IIRC. It’s perfectly possible to machine one though, but as a fairly demanding exercise. Blanks are available commercially.
For your existing 4-jaw, a very useful accessory to help truing a workpiece is a second chuck key to be used in conjunction with the other one. Again a DIY exercise, of turning, drilling and filing. Mild steel is plenty good enough. Practice is also a great help.
PS. I wonder if a Myford backplate could be pressed into service with a spacer ring loctited into the over-large register diameter.