Try it this way:

sub Button1_Click {
    open my $file, '<', "bigfile.txt" or die "$!";

    ## One line at a time
    while( <$file> ) {
        ## skip it if it doesn't contain the text
        next unless m[be]; 

        my @posns;
        ## Find the positions
        push @posns, [ $-[0], $+[ 0 ] ] while m[be]g;

        ## Get the current length
        my $oldLen = $RichText->TextLength;

        ## Move the seletion to the end and insert the line
        $RichText->SetSel( -1, -1 );
        $RichText->ReplaceSel( $_ );

        ## for each position found
        for( @posns ) {
            ## Set the selection to it,
            ## adjusting for the length of the previous contents
            $RichText->SetSel( map $oldLen + $_, @$_ );

            ## And highlight it
            $RichText->SetCharFormat( -color => "#ff0000" );
        }
    }
    close $file;
}
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Ask if anything need clarifying.