There are several variables involved in chuck size. Particularly with a 4 jaw. I have 2 lathes, a 9x19 and a 12x36. I am fortunate that both have the same spindle thread, 1-1/2x8. The issue here is that anything I acquire must (nominally) be usable on both machines.
The 4 jaw that comes with many 9x20s looks like a face plate with woodworking jaws attached.
True, it was large enough to use on both machines, but about useless for holding serious metal work. Grizzly part number
H8048. I wasn't real impressed with it. A far more useful one would be Item# H7605, for a much smaller machine. Only a 3", but at a reasonable price and much more useful. I bought a 6" 4 jaw chuck, a real metalworking chuck, that is usable on both machines. Small enough to be usable on the 9x19 yet large enough for the 12x36. But expensive, even as used.
In addition, I have a 4" and a 5" 3 Jaw and a 5" 4 jaw
scroll chuck, self centering. Useful for what I use it for but not viable as a second chuck. There are some others, but for smaller work requiring absolute closure. Those are not really relevant in this discussion. Grizzly has a number of 4 jaw metal working chucks, but they also are right proud of them. I would say lay back and take your time to find one at a reasonable price that fits your machine.
The big issues to consider are "will it or can be made to" fit the machine and the maximum
outside size. As in can the machine swing it? The rest of the concerns will be the
monkey on your back.
Bill Hudson