How to get to Beedle’s island at night

It's much easier if you use an item with aiming capabilities, like the Slingshot or the Beetle. It's possible to see Beedle before you've picked these up, however, so just come back when you get one. However, if you really want to access the shop, throwing pumpkins or pots seems like it should work (judging from the fact they can trigger switches etc. like shots), but it'll probably be difficult and time consuming.

You get the feathers by catching the Chirri with the Bug Catching Net.

Now I know you said it, catching them is hard, but it's actually easy once you know the secret.

To be able to catch anything that could be caught, you need to sneak up on the target (i.e., walk up to it) and swing your net at it. It's easiest to swing sideways leading with the Wii Remote turned sideways. This applies to anything that is evasive or would otherwise run/fly away.

What I would do is sneak up to it with your net out and touching the target. Once you do that, swipe and it should be yours.

Here's a video demonstrating how you could do it: