you can avoid the oneliner BEGIN block using the Maori-farewell like in perl -M'5; $counter=0' -nle ...

But you can avoid the counter at all:

>perl  -nlE "say sprintf qq(caption%03s.txt),$count++" one.txt two.txt
+ three.txt

caption000.txt
caption001.txt
caption002.txt
[download]

PS you get the filename currently processed in $ARGV so you can play with it:

>perl -nlE "say sprintf q(caption%03s.txt),$1 if $ARGV =~ /.*([\d]+).*
+/" one01.txt one02.txt

caption001.txt
caption002.txt
[download]

PPS note that perl -n and perl -p both skip empty files!

ls -l

-rw-rw-r-- 1 io io  0 feb  2 13:28 one.txt
-rw-rw-r-- 1 io io  0 feb  2 13:28 three.txt
-rw-rw-r-- 1 io io 20 feb  2 13:37 two.txt

perl -nlE 'say "processed $ARGV" if eof' *.txt
processed two.txt
[download]

PPPS after choroba's nice below response I cant resist to update the title and to try it:

 perl -MTie::Scalar -M"5;{package SN; use parent -norequire => 'Tie::S
+tdScalar';sub FETCH {print qq(Opening ${$_[0]}); ${$_[0]}}}; tie $ARG
+V, SN" 
      -nlE "say qq(\tprocessed $ARGV) if eof" 001.txt 002.txt 003.txt

Opening 001.txt
Opening 002.txt
Opening 002.txt
        processed 002.txt
Opening 003.txt
[download]

L*