"I cannot implement the recommendations without losing the functionality."
Well then, Perl::Critic is having the desired effect - steering you away from dodgy areas. :-)
"Are there better ways to get the same output without the sloppy coding I have done?"
Yes, others have pointed you towards PadWalker. That's a much better way for a sub to peek at its caller's variables. Much better in that it actually usually works. (Your current solution works through luck - because the variables you're peeking at are actually file-level.
Here's an example using that.
Better still would be to actually pass the subs the values they need as arguments, so they don't need to peek at their caller's variables...
perl -E'sub Monkey::do{say$_,for@_,do{($monkey=[caller(0)]->[3])=~s{::}{ }and$monkey}}"Monkey say"->Monkey::do'
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