05AB1E, score = 1673 (7 bytes · 239)

žBœ∊{3ý

Try it online!

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 = 2160 (10 bytes · 216)

sm+d+4d.p4

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How it works

 m     .p4   map over all permutations d of [0, 1, 2, 3]:
  +d+4d        d + [4] + d
s            concatenate

05AB1E, score = 1673 (7 bytes · 239)

žBœ∊{3ý

Try it online!

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

05AB1E, 1728 (8 bytes · 216)

žBœvy3yJ

Try it online!

žB          push 1024
  œ         permutations: ["1024", "1042", "1204", …, "4201"]
   v        for each permutation y:
    y         push y
     3        push 3
      y       push y
       J      concatenate the stack

Pyth, score = 2160 (10 bytes · 216)

sm+d+4d.p4

Try it online!

How it works

 m     .p4   map over all permutations d of [0, 1, 2, 3]:
  +d+4d        d + [4] + d
s            concatenate

05AB1E, score = 1673 (7 bytes · 239)

žBœ∊{3ý

Try it online!

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 = 2160 (10 bytes · 216)

sm+d+4d.p4

Try it online!

How it works

 m     .p4   map over all permutations d of [0, 1, 2, 3]:
  +d+4d        d + [4] + d
s            concatenate

05AB1E, score = 1944 (9 bytes · 216)

žBœε3yJ}J

Try it online!

How it works

žBœε3yJ}J
žB          push 1024
  œ         permutations: ["1024", "1042", "1204", …, "4201"]
   ε   }    map over elements y:
    3       push 3
     y      push y again
      J     concatenate the stack
        J   concatenate

Pyth, score = 2160 (10 bytes · 216)

sm+d+4d.p4

Try it online!

How it works

 m     .p4   map over all permutations d of [0, 1, 2, 3]:
  +d+4d        d + [4] + d
s            concatenate

05AB1E, 1728 (8 bytes · 216)

žBœvy3yJ

Try it online!

žB          push 1024
  œ         permutations: ["1024", "1042", "1204", …, "4201"]
   v        for each permutation y:
    y         push y
     3        push 3
      y       push y
       J      concatenate the stack

Pyth, score = 2160 (10 bytes · 216)

sm+d+4d.p4

Try it online!

How it works

 m     .p4   map over all permutations d of [0, 1, 2, 3]:
  +d+4d        d + [4] + d
s            concatenate