I don't think that you will have to repeat the process more than once, so I suggest that you do this manually. Check again if the new points are not close to each other and repeat the selection process if needed until you have 200 points. then you select by location all the points that are within 250 km of your first sample of point, invert the selection and manually select the "x" first points to have a total of 200 points.
Create a buffer around a map point and display the results as a Graphic. On the Insert tab, in the Map Frames group, click Grid. You should therefore build 125 km buffers around each point, with dissolved boundaries, and select by attributes the buffers with an area larger than pi*125000*125000 m²(which means that you have at least two touching buffers) and only keep the point with the smallest random value. How to create Grid in ArcGIS with required Dimension - YouTube On the status bar at the bottom of the active map or scene, click, On the status bar at the bottom of the active map or scene, hover over. The probability that you have two points at less than 250 km from each others is quite small, but it could happen. Once you have the random number, sort your attribute table and take the 200 first points (select manually is the fastest). If you have a set of point, you can random select from it based on a random number in your attribute table.