Simply add the filenames as a new field:
$ cut -f 2 HNs.bst_.lt_.txt | sort > HNusage/HNs.bst.lt/temp
$ awk '{ print $2, $1, FILENAME }' *.txt | sort -k 1,1 | join - HNusage/HNs.bst.lt/temp

In the script:
#!/bin/sh
if [ -z "$2" ]
then
    printf "Usage: $0 searched-domain-file file1 [file2 ...]
"
    exit
fi
searched="$1"
shift
TMP=$(mktemp)
trap "rm $TMP 2>/dev/null" 0
mkfifo $TMP
cut -f 2 "$searched" | sort > $TMP &
awk '{ print $2, $1, FILENAME }' "$@" | sort -k 1,1 | join - $TMP

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