Match this random string To-Do: Come up with a Better Title

Specification:
This challenge is similar to kolmogorov-complexity, but with a twist. You goal is to output a finite string that resembles the following string of psuedo-random characters as closely as possible:
BABBBBAAABBAABBBBAABBBAABBBABBAABABABBBBABBBBBBAABABBAABABBBABBBABAAAAABAAAABBAAABAAABBBBAABBABBBAABBBAABAAAABAAAABABABBBBBAABAAABBABAABAABABABBAABAAABBBABBABABBABAABABABBAABBBAABABABBBABAAABAABBBBBAABBBABABAABAABBAABABBBAABBAABAAABABABBBBBAAAAABBBAAABABAABBAABAABAAAABABBBAABBABBBBBAABAAAAABABBBAAAAAABAAAABBABAABABBAAABBBBBBAABABBAAAAAABABBAAABBABBABBBBAAAAAABABBBBBABBBABBAABBABAABAAAAABBAABABBABAABABABBBAAAAAABBBBBABABBBBBAABAABBBAAABABBABBAAABBABBABABBAABBAABAABBBABBABABBABABABABABBBAAABBBBBBABAAAAAABBAABBAABBBAAAAAAABBBBABBBBAABAABAABBABBAAAAAABABBABAAAAABABABAABAABBBAABBABAAABABAAAAABAABABAABBABAABBABBABAABABBBBAAAAAABAAAAAAAAABAABAAAAAAAABABAABBBABAAABABBBBBAAABAAAABBBBBABBBAAAAABBBABBAAAABAABBBBAABBABABAAABAABAAAAABBABBBABBBBAABBABABBAABAAAABAABBABABBBAABABABABAABAAAAABBABBBABBABBAAABAAAABBABABBBABBABABBBBABAAABBBBAAAABBABAAABABBBBBABABBAABBABAAABBABAAAAAABBBAAAABBBAAABABAAAAABBBBABAAABBBAAAABABBBBAAAAABBBBAABAABABBABABBABAAAABBBBAAABAABAAABBBBBBABBABAABABAABBBBBBBBAABBBAABAABABAABBAAABABAABAABBBBBBABBA

The string is 1024 characters long, and consists only of uppercase A and B characters. You must write a program takes up less than or equal to 256 bytes. The above string was generated completely randomly.
This challenge is not code-golf! Since 256 bytes is far too few characters (even for golfing languages), you may output any finite string. Your score is Levenshtein Distance between the 1024 byte string, and the string your program outputs (where the lower score is the best.)

Examples:
Input, Output
1024 &quot;A&quot; characters, 500 Points
1024 &quot;B&quot; characters, 524 Points
1024 characters alternating between &quot;A&quot; and &quot;B&quot; (&quot;A&quot; first), 348 Points
1024 characters alternating between &quot;A&quot; and &quot;B&quot; (&quot;B&quot; first), 346 Points


Notes:

The output must be finite and deterministic.

Unfortunately, this challenge is limited to languages that can be scored in bytes.

You may not substitute A and B for other characters. If your language cannot output such characters, it may not compete.

If a language cannot compete, it is still welcome to post non-competing answers. Just make sure to explain how you are scoring the challenge.

The winner of this challenge is on a per-language basis, since would be unlikely that an answer from Brainf*** would ever beat one from Jelly.
PHP, 246 printable bytes, distance 0
while($c=ord(base64_decode(&quot;vGec7K9+WXdBDEebnIQr5GkrI7WlZyuifOpMuZFfBxTJC5vkFwIaWPywLG3gX3ZpBlpXA+vk4tjazJ2tVcfoGZwHvJNgWgqTmiglNNpeBAEgFOi+IfcHYTzUSDd5rITXKpBu2Ia7Xo8NF9Zo0DhxQejheDyWtDxI/aU/zkpik/Y&quot;)[$i++]))echo strtr(sprintf(&quot;%08b&quot;,$c),10,BA);

or
while($c=base64_decode(&quot;vGec7K9+WXdBDEebnIQr5GkrI7WlZyuifOpMuZFfBxTJC5vkFwIaWPywLG3gX3ZpBlpXA+vk4tjazJ2tVcfoGZwHvJNgWgqTmiglNNpeBAEgFOi+IfcHYTzUSDd5rITXKpBu2Ia7Xo8NF9Zo0DhxQejheDyWtDxI/aU/zkpik/Y&quot;)[$i++])echo strtr(sprintf(&quot;%08b&quot;,ord($c)),10,BA);

Remarkable: gnuzip fails on the binary string. gzcompress adds 11 bytes and gzdeflate adds 5.
The fact that no 0x00 byte is in there saves 2 bytes on the first version;
no 0x30 saves 2 bytes on the second one.