Despite custom dimensions having weather independent of the Overworld, it can't be controlled by any commands, even when using /execute. This should affect any datapack with a dimension that uses the "Overworld" settings for sky and fog.

Steps to reproduce:
1. In the Overworld, run /execute in minecraft:overworld run weather rain
The weather will switch to rain, as expected.
2. Teleport to your custom dimension, then run /execute in [your dimension] run weather rain
The weather will not change.

Strangely enough, just because you can't control the weather in a custom dimension does not mean it doesn't have any. If you sit long enough in a custom dimension, rain and thunder can appear like they may in the Overworld, you just can't control it.