Wolfram Language (Mathematica), 263 bytes, score ~150000 on TIO

With[{size = 2 Ceiling[Sqrt@#/2.7] + 3}, 
   Print @@ 
    FixedPoint[
     BitAnd[#, 3] + 
       With[{q = BitShiftRight[#, 2]}, 
        Sum[RotateLeft[q, d], {d, {1, -1, size, -size}}]] &, 
     CenterArray[#, size^2]]] &@ToExpression[$ScriptCommandLine[[2]]]

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Wolfram Language (Mathematica), 284 bytes, score ~150000 on TIO

Fixed a bug thanks to @att.

With[{size = 2 Ceiling[Sqrt@#/2.7] + 3}, 
   WriteString[$Output, ##] & @@ 
    FixedPoint[
     BitAnd[#, 3] + 
       With[{q = BitShiftRight[#, 2]}, 
        Sum[RotateLeft[q, d], {d, {1, -1, size, -size}}]] &, 
     CenterArray[#, size^2]]] &@ToExpression[$ScriptCommandLine[[2]]]

Try it online!