You need to tell it to read the file all at once instead of line-by-line. Then you need to adjust your regex to do multiline matching (and not to be so greedy as to eat the numbers):

perl -0777 -ne 'if (m/^Total SE ON.*?(\d+)/ms) {print "$1\n"}' *
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Update: as ikegami notes, paragraph mode would also be suitable. The shorthand for that is -00 (those are zeroes). See perlrun