try using a script

.bumpversion.toml

@@ -2,7 +2,6 @@
 current_version = "1.3.5"
 commit = true
 tag = true
-push = true
 tag_name = "{new_version}"
 
 [[tool.bumpversion.files]]

bump_and_push.sh

@@ -0,0 +1,18 @@
+#!/bin/bash
+set -ex
+
+# Check if an argument is provided
+if [ -z "$1" ]; then
+  echo "Usage: $0 <part>"
+  echo "Example: $0 patch"
+  exit 1
+fi
+
+PART=$1
+
+# Bump the version
+bump-my-version bump $PART
+
+# Push the changes
+git push
+git push origin --tags