Your solution which adds R? to the original regex was on the right track and achieves what you want, albeit poorly:

$ perl -Mstrict -Mwarnings -E 'my $fileName;
my $re = qr{3B4[0|1|2]RT\.\d{10}\.\d+R?\.bin};
$fileName = "3B40RT.2000033121.7.bin.gz";
say +($fileName =~ /$re/) ? "match" : "no match";
$fileName = "3B40RT.2000033121.7R.bin.gz";
say +($fileName =~ /$re/) ? "match" : "no match";
'
match
match
[download]

I don't think you understand character classes or alternation (perhaps both). Where you're trying to match a 0, 1 or 2 in the same position, [0-2] would be far better than [0|1|2] (which is trying to match a 0, pipe, 1, pipe or 2 in the same position) - the 2nd pipe is redundant and the 1st pipe isn't wanted anyway. So, here's an improved version:

$ perl -Mstrict -Mwarnings -E 'my $fileName;
my $re = qr{3B4[0-2]RT\.\d{10}\.\d+R?\.bin};
$fileName = "3B40RT.2000033121.7.bin.gz";
say +($fileName =~ /$re/) ? "match" : "no match";
$fileName = "3B40RT.2000033121.7R.bin.gz";
say +($fileName =~ /$re/) ? "match" : "no match";
'
match
match
[download]

Recommended reading:

• perlrequick - Perl regular expressions quick start
• perlretut - Perl regular expressions tutorial
• perlre - Perl regular expressions, the rest of the story
• Writeup Formatting Tips