Switch keyboard profile programmatically

This piece of code will fulfill your requirements:

InputMethodManager imeManager = (InputMethodManager) getApplicationContext().getSystemService(INPUT_METHOD_SERVICE);
imeManager.showInputMethodPicker();

As Commonsware points out in his answer, there is no way to do this behind the user's back.

If your app has system privileges, and has the permission

<uses-permission android:name="android.permission.WRITE_SECURE_SETTINGS" />

you can programatically enable the keyboard and set it as the current keyboard by making it the default keyboard WITHOUT USER KNOWLEDGE OR INTERVENTION!

//get the old default keyboard in case you want to use it later, or keep it enabled
String oldDefaultKeyboard = Settings.Secure.getString(resolver, Setting.Secure.DEFAULT_INPUT_METHOD);

//enable your keyboard
Settings.Secure.putString(resolver, Settings.Secure.ENABLED_INPUT_METHODS, "com.my.keyboard/.full.path");

//set your keyboard as the new default keyboard
Settings.Secure.putString(resolver, Settings.Secure.DEFAULT_INPUT_METHOD, "com.my.keyboard/.full.path");

You can enable multiple keyboards (such as the default keyboard and your own) by providing a list of keyboards to the ENABLED_INPUT_METHODS, separated by ':'. See docs

You can verify your keyboard's full package and path ID by invoking ime list -a through adb shell