It may be easier to draw a small section of the staggered polygon around just one of the LEDs that's aligned to the rectangular grid and then copy, paste, and rotate it around to each additional LED. Optionally combine all of them into one polygon at the end.
Altium's handling of arc is pretty annoying, especially in polygon outlines, so you may get better results by using the arc and line primitives to draw the outline you want and then convert to a polygon from there. I would probably be tempted to export the LED layout to DWG, draw the polygons in AutoCAD or inkscape or whatever, and then import the DWG back into the PCB as primitives and then convert to a polygon.