resource_group client: narrow degraded fallback conditions for GetResourceGroup

## Enhancement Task

As suggested in PR comment: https://github.com/pingcap/tidb/pull/68105#discussion_r3584191593

Refine the degraded fallback behavior of `GetResourceGroup` when `degradedRUSettings` is configured.

Currently, degraded fallback is too broad and may hide original errors from RM. We should only use degraded fallback for transient RM failures, while preserving original errors for non-retryable cases.

Expected behavior:
- transient RM unavailability -> degraded group
- RM recovery -> real group
- resource group not found -> original error
- caller cancellation/deadline -> original context error
- generic non-retryable error -> original error
- nonexistent `SwitchGroup` target -> do not switch

It is also helpful to preserve the underlying error from `ErrClientGetResourceGroup` so upper layers can correctly inspect errors such as `context.Canceled` and `context.DeadlineExceeded`.

Start by tracing the GetResourceGroup path and how degradedRUSettings is applied, then inspect ErrClientGetResourceGroup and SwitchGroup handling. Verify each listed outcome: transient RM failures use the degraded group, recovery uses the real group, and not-found, context, generic non-retryable, and nonexistent-target cases preserve the original behavior and errors.

Written by the indexing model from the issue text.