I am having a hard time understanding the problem. You write, "no established list of all the possibilities". I looked at the JSON output of https://api.weather.gov/icons. That looks like the possibilities to me. I decoded the JSON and simplified that into a more straightforward translation table. I do not show the LWP code, but I guess you know how to do that.

In general for things like this, I have found that using the textual description on the website is better than "rolling your own". If say, you need a different text for tsra_sct than the website provides, perhaps you want "tsra_sct" and "tsra_hi" to translate into the same thing? Then we are into a different discussion about how to maintain such a thing. That is a significant ongoing hassle that I don't recommend.

This gets the textual defs of each abbreviation from the website and translates the first "word" argument of the last path in the URL to that textual definition. I translated each of the URL's you provided. Please explain what else you need...

#!/usr/bin/perl
use strict;
use warnings;

use JSON::Parse 'parse_json';


my $json = do{local $/ = undef;<DATA>};

my $out = parse_json $json;
my %xlated_abbrev;  #simple abbreviation table => description

foreach my $key (keys %{$out->{icons}}) #gen simple xlate table
{
   $xlated_abbrev{$key} = $out->{icons}{$key}{description};
}

my @urls = (
'https://api.weather.gov/icons/land/day/tsra_sct,20/tsra_sct,40?size=m
+edium',
'https://api.weather.gov/icons/land/day/rain_showers,30/tsra_hi,30?siz
+e=medium',
'https://api.weather.gov/icons/land/night/rain_showers,30/rain_showers
+?size=medium',
'https://api.weather.gov/icons/land/day/bkn?size=medium'
);

foreach my $url (@urls)
{
   my $last_path = (split('/',$url))[-1];
   my ($abbrev_to_xlate) = $last_path =~ /^(\w+)/;

   print "URL = $url\n";
   print "   $abbrev_to_xlate => \'$xlated_abbrev{$abbrev_to_xlate}\'\
+n\n";
}

=PRINTS:
URL = https://api.weather.gov/icons/land/day/tsra_sct,20/tsra_sct,40?s
+ize=medium
   tsra_sct => 'Thunderstorm (medium cloud cover)'

URL = https://api.weather.gov/icons/land/day/rain_showers,30/tsra_hi,3
+0?size=medium
   tsra_hi => 'Thunderstorm (low cloud cover)'

URL = https://api.weather.gov/icons/land/night/rain_showers,30/rain_sh
+owers?size=medium
   rain_showers => 'Rain showers (high cloud cover)'

URL = https://api.weather.gov/icons/land/day/bkn?size=medium
   bkn => 'Mostly cloudy'
=cut



#Data returned from: https://api.weather.gov/icons
__DATA__
{
    "@context": [],
    "icons": {
        "skc": {
            "description": "Fair/clear"
        },
        "few": {
            "description": "A few clouds"
        },
        "sct": {
            "description": "Partly cloudy"
        },
        "bkn": {
            "description": "Mostly cloudy"
        },
        "ovc": {
            "description": "Overcast"
        },
        "wind_skc": {
            "description": "Fair/clear and windy"
        },
        "wind_few": {
            "description": "A few clouds and windy"
        },
        "wind_sct": {
            "description": "Partly cloudy and windy"
        },
        "wind_bkn": {
            "description": "Mostly cloudy and windy"
        },
        "wind_ovc": {
            "description": "Overcast and windy"
        },
        "snow": {
            "description": "Snow"
        },
        "rain_snow": {
            "description": "Rain/snow"
        },
        "rain_sleet": {
            "description": "Rain/sleet"
        },
        "snow_sleet": {
            "description": "Rain/sleet"
        },
        "fzra": {
            "description": "Freezing rain"
        },
        "rain_fzra": {
            "description": "Rain/freezing rain"
        },
        "snow_fzra": {
            "description": "Freezing rain/snow"
        },
        "sleet": {
            "description": "Sleet"
        },
        "rain": {
            "description": "Rain"
        },
        "rain_showers": {
            "description": "Rain showers (high cloud cover)"
        },
        "rain_showers_hi": {
            "description": "Rain showers (low cloud cover)"
        },
        "tsra": {
            "description": "Thunderstorm (high cloud cover)"
        },
        "tsra_sct": {
            "description": "Thunderstorm (medium cloud cover)"
        },
        "tsra_hi": {
            "description": "Thunderstorm (low cloud cover)"
        },
        "tornado": {
            "description": "Tornado"
        },
        "hurricane": {
            "description": "Hurricane conditions"
        },
        "tropical_storm": {
            "description": "Tropical storm conditions"
        },
        "dust": {
            "description": "Dust"
        },
        "smoke": {
            "description": "Smoke"
        },
        "haze": {
            "description": "Haze"
        },
        "hot": {
            "description": "Hot"
        },
        "cold": {
            "description": "Cold"
        },
        "blizzard": {
            "description": "Blizzard"
        },
        "fog": {
            "description": "Fog/mist"
        }
    }
}
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