I'd go for setting $/ (input_record_separator) so that you read a whole logical record at a time; that makes the numbering and editing a lot easier:

{
    local $/ = "}\n";
    while (<>) {
        s/^"\d+"/"$."/;
        print;
    }
}
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(This is based on your later clarification that the records themselves do not need to be re-ordered according to the original record numbers before assigning the new sequential record numbers.)