Hi! I am currently residing in Bergamo, and in the middle of March, I am traveling by car to Pila and perhaps even Courmayeur. I am checking accommodation in advance, but everything already seems so expensive both on Booking and AirBnb! Do you have any suggestions on where to stay in Pila or 10km around and not to break the bank?

You can stay in Aosta. It is actually a big city at 500 m over the sea level but maybe you can find more cheap hotel

The average price of an apartment in Pila this evening is $146 (based on Booking.com prices). I recommend Silver Apartments, it has self-catering accommodations with a balcony and free WiFi. Show more Show less from $58 per night.
duck life

Remain in Aosta, please. At 500 meters above sea level, it's a fairly large city, but perhaps there are more affordable hotels there.