Improve `LightClientStore` update after sync committee change proof generation

In the Client for the Ethereum Light Client, we currently update the `LightClientStore` with `process_light_client_update` once the proof has been generated and verified.

By having better condition on the received update to check which filed should be updated, we could have a more clever way to update the store without having to run through the whole logic again.

Start with the LightClientStore update path around process_light_client_update and trace how the store is changed after an update is generated and verified. Determine which received-update condition identifies the field affected by a sync committee change, then confirm that the store can be updated without rerunning the full processing logic.

Written by the indexing model from the issue text.