If your thinking of a replacement engine.... before I would do that, I would just add oil and ride it.
If the rear wheel locked up while riding, it probably was the pistons seizing in the cylinder. The cams would not lock it up and the transmission doesn't need much oil.
Unless you remove the plug, you will not be able to push start it unless your in 2nd or 3rd.
If your going to try and rebuild it, I would check the cylinder/pistons/cam lobes. Check the oil pump or replace.

 

Do a compression test if you haven't already.

 

Wouldn't any KLX engine fit in there? Would a KLX300R engine fit in there?

 

The 300 might fit right up but the electrical wouldn't match up. If you could put the 250 stator, CDI, and...well the whole 250 electrical system in the 300 it might work. You'd also have to use the left side cases from the 250 to keep the starter.
A 300 motor could be less expensive and have a lot less miles on it, maybe.