I had a charging problem with my first Apple Pencil. I called a support number, maybe the one for iPads, and the customer support person wanted the serial number. It might have been on the box but I couldn't locate the box as I probably recycled it. Since I bought my pencil from Apple online I was able to download a copy of the invoice receipt, which has a product number and a serial number for the pencil. That was enough information.
The customer support person wanted me to go to a store but even though they're not as far away from me as you it was very inconvenient at that time so I insisted that an item purchased online should be exchangeable online. That required a supervisor but it was done. I was instructed to return the broken pencil in the packing material the replacement arrived in, with an included prepaid label, so it was a very smooth process.
The downside of doing it this way is that I received a pencil that had been returned w/o the extra tip or charging doodad. Now this pencil drops the Bluetooth connection almost very day but I'm going to try to make the exchange in a store so I get a new one.