[model-gateway] Optimize memory usage in HTTP router

[slin1237]

Key optimizations:

1. tree.rs: Fix O(n²) character iteration

   • Add CharIndexedText struct for O(1) character access
   • Replace chars().nth(idx) with pre-indexed Vec<char>
   • Use shared_prefix_count_indexed() to avoid intermediate strings

2. pd_router.rs: Reduce allocations

   • Use Arc to share request across retry attempts (avoids O(retries) clones)
   • Use static string constants for bootstrap JSON keys
   • Optimize logprob merging with std::mem::take to avoid double cloning
   • Pre-allocate merged arrays with exact capacity

3. router.rs: Reduce string allocations

   • Pre-allocate Bearer token header with capacity
   • Use static string constant for DP rank key

4. chat.rs: Optimize to_simple_string()

   • Build string directly without intermediate Vec allocation

Checklist

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