Run NEAR tool on itself and select points with near distance greater or equal to 250 km.
If you are lucky to get 200 points or more, copy them to a separate feature class and use any random generator to pick 200 of them. Selected points in a picture below show points sitting at least 500 m away from any other point.
If there are less than 200 points, ...
Given you already have your RED 6km span points which I am referring to red_points, you can do something like this to obtain your 3X3 200m grey_points.
(please, adjust the sign (+/-) as you wish to achieve your result. Also, here 200 is in meters as long as the units of your reference system is meter.)
red_points = "your_red_points_fc_here"
If you have a set of point, you can random select from it based on a random number in your attribute table. This is the first step. Once you have the random number, sort your attribute table and take the 200 first points (select manually is the fastest).
The probability that you have two points at less than 250 km from each others is quite small, but it ...