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/Migrations/20260606185411_AddLitterLetter.cs

@@ -0,0 +1,28 @@
+using Microsoft.EntityFrameworkCore.Migrations;
+
+#nullable disable
+
+namespace GerbilManagerWebAPI.Migrations
+{
+    /// <inheritdoc />
+    public partial class AddLitterLetter : Migration
+    {
+        /// <inheritdoc />
+        protected override void Up(MigrationBuilder migrationBuilder)
+        {
+            migrationBuilder.AddColumn<string>(
+                name: "LitterLetter",
+                table: "Litters",
+                type: "text",
+                nullable: true);
+        }
+
+        /// <inheritdoc />
+        protected override void Down(MigrationBuilder migrationBuilder)
+        {
+            migrationBuilder.DropColumn(
+                name: "LitterLetter",
+                table: "Litters");
+        }
+    }
+}

GerbilManagerWebAPI/Migrations/ApplicationContextModelSnapshot.cs

@@ -905,6 +905,9 @@ namespace GerbilManagerWebAPI.Migrations
                     b.Property<Guid?>("FatherId")
                         .HasColumnType("uuid");
 
+                    b.Property<string>("LitterLetter")
+                        .HasColumnType("text");
+
                     b.Property<Guid?>("MotherId")
                         .HasColumnType("uuid");