fix: use node:20-alpine in sync builder, matching backend Dockerfile

node:20-slim picks up the Apple npm proxy which blocks installs.
node:20-alpine does not. Also add PATH for node_modules/.bin so tsc
is found, and remove committed dist/ since the build now works properly.

Co-Authored-By: Claude Sonnet 4.6 <noreply@anthropic.com>

sync/Dockerfile

@@ -1,11 +1,20 @@
-# dist/ is compiled locally and committed to the repo.
-# The server npm proxy blocks devDependency installs inside Docker.
+# Stage 1: build TypeScript
+FROM node:20-alpine AS builder
+WORKDIR /app
+RUN apk add --no-cache python3 make g++
+COPY package*.json ./
+RUN npm ci --include=dev
+ENV PATH="/app/node_modules/.bin:$PATH"
+COPY tsconfig.json ./
+COPY src ./src
+RUN npm run build && npm prune --production
+
+# Stage 2: runtime with Playwright
 FROM mcr.microsoft.com/playwright:v1.47.0-jammy
 WORKDIR /app
 RUN npx playwright install chromium --with-deps
-COPY package.json package-lock.json ./
-RUN npm install --omit=dev --registry https://registry.npmjs.org
-COPY dist ./dist
+COPY --from=builder /app/node_modules ./node_modules
+COPY --from=builder /app/dist ./dist
 
 CMD ["node", "dist/index.js"]