Characters with skins from custom skin packs fail to load when restarting game

Steps to Reproduce:

1. Download and import the attached skin pack
2. Open the Dressing Room and create a character
3. Equip the skin from the custom skin pack
4. Create a second character
5. Exit the Dressing Room with the second character selected
6. Restart the game
7. Open the Dressing Room
8. Select the character with the custom skin pack skin

Observed Results:
The character fails to load. Pressing the "Reload" button only shows the "Failed to load character" message.

Expected Results:
Characters with skins from custom skin packs should successfully load when restarting the game, just as they did before 1.19.0.24.

Notes:
One way to get the character to load is to select the second character, open the classic skins menu, back out, and then press the "Reload" button.

Video:
Minecraft 2022-04-17 15-11-21.mp4