05AB1E, score = 1673 (7 bytes · 239)

žBœ∊{3ý

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How it works

žB          push 1024
  œ         permutations: ["1024", "1042", …, "4201"]
   ∊        vertically mirror: ["1024", "1042", …, "4201", "4201", …, "1042", "1024"]
    {       sort: ["0124", "0124", "0142", "0142", …, "4210", "4210"]
     3      push 3
      ý     join: "01243012430142301423…3421034210"

Pyth, score = 1944 (9 bytes · 216)

s+R+4d.p4

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How it works

 +R   .p4   append to each permutation d of [0, 1, 2, 3]:
   +4d        [4] + d
s           concatenate