feat: implement M3 multi-browser support (Firefox, Edge, Safari)

Add browser abstraction layer (compat.ts) that normalizes Chrome/Firefox/
Edge/Safari extension APIs behind a unified promise-based interface.
Replace all direct chrome.* calls with compat layer across service worker,
sync engine, badge, storage, popup, and options modules.

- Browser-specific manifests in manifests/ (Firefox MV3 with gecko settings,
  Edge/Safari variants)
- Multi-target build system: `npm run build` produces all four browser
  builds in build/<browser>/
- Per-browser build scripts: build:chrome, build:firefox, build:edge, build:safari
- Auto-detects browser at runtime for platform-specific device registration
- All 38 existing tests pass, typecheck clean

Co-Authored-By: Paperclip <noreply@paperclip.ing>

extension/src/popup/popup.ts

@@ -1,6 +1,9 @@
 /**
  * Popup script — controls the popup UI interactions.
+ * Uses the browser-agnostic compat layer.
  */
+import { runtime, tabs } from "../lib/compat";
+
 export {};
 
 // --- Elements ---
@@ -40,20 +43,12 @@ const btnPairingCancel = document.getElementById("btn-pairing-cancel") as HTMLBu
 
 // --- Messaging helper ---
 
-function sendMessage(type: string, payload?: unknown): Promise<any> {
-  return new Promise((resolve, reject) => {
-    chrome.runtime.sendMessage({ type, payload }, (response) => {
-      if (chrome.runtime.lastError) {
-        reject(new Error(chrome.runtime.lastError.message));
-        return;
-      }
-      if (response?.error) {
-        reject(new Error(response.error));
-        return;
-      }
-      resolve(response);
-    });
-  });
+async function sendMessage(type: string, payload?: unknown): Promise<any> {
+  const response = await runtime.sendMessage({ type, payload });
+  if (response?.error) {
+    throw new Error(response.error);
+  }
+  return response;
 }
 
 // --- UI Updates ---
@@ -142,7 +137,7 @@ btnSyncTab.addEventListener("click", async () => {
 });
 
 btnAddWhitelist.addEventListener("click", async () => {
-  const [tab] = await chrome.tabs.query({ active: true, currentWindow: true });
+  const [tab] = await tabs.query({ active: true, currentWindow: true });
   if (!tab?.url) return;
 
   try {
@@ -198,7 +193,7 @@ btnPairingCancel.addEventListener("click", () => {
 
 btnSettings.addEventListener("click", (e) => {
   e.preventDefault();
-  chrome.runtime.openOptionsPage();
+  runtime.openOptionsPage();
 });
 
 // --- Init ---