I went with

next if ($chromosome =~ "chrM");

That works, but personally I wouldn't write it that way, because writing a regex like for example /chrM/ or m{chrM} makes it more visually clear what you want to do (and also allows you to add modifiers).

I was using both Strict and Warnings and using the quotes instead of the regex doesn't produce any warnings

Are you sure? next if ($chromosome = "chrM"); should have given you the warning "Found = in conditional, should be ==". Perhaps you're not enabling warnings correctly?

Update:

My apologies about the input data format

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