power supply - Prevent overcurrent while filter capacitor is charged

I have this circuit:

On start-up, the power supply experiences an overcurrent condition, because when the capacitor is charged, it looks like a short circuit as seen from the power supply. How can I prevent this?

Adding a current-limiting resistor before the capacitor is not suitable for me, because of the load-dependent voltage drop - the load, of course, will look smaller once the capacitor has charged at the maximum current of about 2 A. Maybe there is a simple circuit to temporarily enable a current-limiting resistor? Or another simple solution?

ADDED. I have these ideas. Please feel free to comment.