Walkthroughs for custom maps.

A very strange army composition, but it seems that you need to defeat them specifically with troglodytes.

That's right, specifically with troglodytes. The tactic turned out to be quite simple.
Turn 1
1. On the first turn, the computer should cast Counterspell instead of Slow.
2. Wait for the Rust Monsters and Archmages.
3. Attack the Air Elementals "head-on" with the Rust Monsters, and attack diagonally with the Archmages.
4. The Phoenixes fly up to the Archmages, but don't attack (!), while the Water Elementals end up under attack from the Phoenixes through one cell.
5. Attack the Water Elementals with the Fanatics.
You can skip moving the other troops, but it's better to move the troglodytes closer to the center.
Turn 2
1. I don't know what the Phoenixes are doing, I don't remember, they seem to be waiting.
2. Attack the Air Elementals with the Archmages.
3. Attack the Phoenixes with the Azure Dragons in such a way that they retaliate and hit their own Water Elementals, while the Azure Dragons block the passage to the Archmages for the Magic Elementals.

If the Archmages are morale-broken during these 2 turns, the Air Elementals will die (3 hits from the Archmages + 2 retaliations). Then, you need to attack the Phoenixes with the Archmages in the same way as the Azure Dragons, so that they hit the Water Elementals. As a result, about 100 Water Elementals remain, which the computer somehow uses to exit the city, blocking the gates.
After that, everything is simple - the Fire Elementals and Magic Elementals are blocked in the city, the troglodytes attack the Energy Elementals and Phoenixes, and the other troops don't matter. You just need to remember to cast Dispel on the enemy units if the computer casts Shield or Hide. By the time the catapult breaks through the walls and the Fire Elementals and Magic Elementals come out of the city, the troglodytes will have already destroyed the Phoenixes and Energy Elementals. The same fate awaits those who are late. In the end, about 3k troglodytes remain.