No, I do not advise using different names in the sub from the ones used in the main part of the script. The whole point of lexically scoped names is that a name in one scope does not interfere with a name in a different scope. The programmer can stop worrying about where else a name may have been used, because it doesn't matter.

A note about scope

Instead of

my $day;

foreach $day(@ALL_WEEK_DAYS){
    ...
}
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you should declare your loop variable as part of the foreach statement:

foreach my $day(@ALL_WEEK_DAYS) {
    ...
}
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