I believe you want this:

$ARRGH =~
  s#<doct>Customer trans\.(?:.*)+?</doct>#<doct>Customer trans\.</doct
+>#g;
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What the +? does is makes the match less "greedy", so that it matches the first </doct> it finds instead of the last. Also, since you aren't using what you capture in (.*), I put the ?: in, so that perl doesn't try to retain what it found, but still keeps the grouping.

I also changed the regex delimiter to | so that you don't have to escape the / in </doct>.

Update: As per Aristotle's reply, I've changed the regex delimiter to #.