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One method would be to split your point file (via a select and export or some other method) into to new files - points within buffers and points that are not. From there you can use a Spatial Join (which doesn't require an Advanced License if you don't have one) to join the points outside the buffers to those inside. With the settings of that tool you can ...
Try using the Near (analysis) tool. The resulting table will show which point is closest by whatever OBJECTID you specify. Then perform a join and field calculate the values into your previous points.
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