Here's a couple tips:

UPS has several levels of oversize charges. Some bike boxes may be large enough to trigger the higer cost levels. It helps to check the limits before selecting your box.

You can avoid any Mailbox Etc. / UPS Store upcharges by setting up your own MyUPS account at ups.com. From there, you can create your own shipment, get the price, and print your shipping label. Once you've done that, you can drop your package off at any UPS shipping location, including a UPS Store without paying them anything. By using MyUPS you can enter your package dimensions and see any oversize charges before you print the label.