FEAT-NAMEGEN: GET /names/suggest backend (Gemini, 503 NamesKeyMissing, LitterLetter)

- NameSuggestionService: Gemini via existing OpenAiChatClient, builds
  letter/gender/usages/count prompt, strips markdown fences from response,
  deserialises [{name,meaning,origin}] array; NotConfigured -> 503.
- GET /names/suggest?letter=&gender=&usages=&count= -> Ok<List<NameSuggestion>>
  | 503 {code:NamesKeyMissing} | 502 {code:NamesUpstreamError}.
- Litter.LitterLetter (string?, nullable) + AddLitterLetter migration.
- 17 new tests (prompt assembly, fence strip, parse edge cases,
  503-not-configured, upstream-error); total 157/157 green.
- No Behind-the-Name dependency — Gemini path only (Julian's decision).

Co-Authored-By: Claude Sonnet 4.6 (1M context) <noreply@anthropic.com>

GerbilManagerWebAPI/Models/Litter.cs

@@ -30,5 +30,9 @@ namespace GerbilManagerWebAPI.Models
         /// nulls allowed for manually-entered litters). Primary idempotency key for re-imports;
         /// Name+Date is the fallback for litters created before this column existed.</summary>
         public string? ExternalRef { get; set; }
+
+        /// <summary>FEAT-NAMEGEN: Wurfbuchstabe (A, B, C … AA, AB …) — alle Welpen dieses
+        /// Wurfs erhalten Namen mit diesem Anfangsbuchstaben (gängige Zuchtkonvention).</summary>
+        public string? LitterLetter { get; set; }
     }
 }