[docker] Bundle device-builder 1.0.1, make HA add-on builder-only (#16989)

Co-authored-by: J. Nick Koston <nick@koston.org>

docker/Dockerfile

@@ -1,5 +1,5 @@
 ARG BUILD_VERSION=dev
-ARG BUILD_BASE_VERSION=2025.04.0
+ARG BUILD_BASE_VERSION=2026.06.0
 ARG BUILD_TYPE=docker
 
 FROM ghcr.io/esphome/docker-base:debian-${BUILD_BASE_VERSION} AS base-source-docker
@@ -31,6 +31,9 @@ RUN \
     uv pip install --no-cache-dir \
     -r /requirements.txt
 
+# Install the ESPHome Device Builder dashboard.
+RUN uv pip install --no-cache-dir esphome-device-builder==1.0.1
+
 RUN \
     platformio settings set enable_telemetry No \
     && platformio settings set check_platformio_interval 1000000 \