I have not tested this but I think what I would try is:

1. Keep a copy of your original point feature class
2. Run Point Distance on your feature class so that you can determine the point which has the most other points within 400 metres of it.
3. Delete that point and repeat step 2, and keep repeating until you have no points with any points within 400 metres.
4. Buffer those points by 400 metres and draw the original points on the buffer polygons.

If you see any points outside the buffers reassess this method.

Point Distance requires an Advanced license which you do not have, but there is an available workaround for Performing Point Distance analysis using Basic level license of ArcGIS Desktop?

I have not tested this but I think what I would try is:

1. Keep a copy of your original point feature class
2. Run Point Distance on your feature class so that you can determine the point which has the most other points within 400 metres of it.
3. Delete that point and repeat step 2, and keep repeating until you have no points with any points within 400 metres.
4. Buffer those points by 400 metres and draw the original points on the buffer polygons.

If you see any points outside the buffers reassess this method.

Point Distance requires an Advanced license which you do not have, but there is an available workaround for Performing Point Distance analysis using Basic level license of ArcGIS Desktop?

I have not tested this but I think what I would try is:

1. Keep a copy of your original point feature class
2. Run Point Distance on your feature class so that you can determine the point which has the most other points within 400 metres of it.
3. Delete that point and repeat step 2, and keep repeating until you have no points with any points within 400 metres.
4. Buffer those points by 400 metres and draw the original points on the buffer polygons.

If you see any points outside the buffers or any buffers that overlap then reassess this method.

Point Distance requires an Advanced license which you do not have, but there is an available workaround for Performing Point Distance analysis using Basic level license of ArcGIS Desktop?

I have not tested this but I think what I would try is:

1. Keep a copy of your original point feature class
2. Run Point Distance on your feature class so that you can determine the point which has the most other points within 400 metres of it.
3. Delete that point and repeat step 2, and keep repeating until you have no points with any points within 400 metres.
4. Buffer those points by 400 metres and draw the original points on the buffer polygons.

If you see any points outside the buffers reassess this method.

Point Distance requires an Advanced license which you do not have, but there is an available workaround for Performing Point Distance analysis using Basic level license of ArcGIS Desktop?