Weight-based Priority Mail is the best way (although not always cheapest way) to ship packages that are too big for Priority Mail Cubic.
You don't have to use USPS-branded boxes, but you can use any of the "Mailing Boxes" provided by USPS: https://support.pirateship.com/usps-shipping-services/do-i-need-to-use-a-priority-mail-box-if-i-buy-a-priority-mail-label
In some cases weight-based Priority Mail rates will be cheaper than dimension-based Priority Mail Cubic rates. In those cases Pirate Ship gives the customer whichever one is cheaper, which is why it’s important to always be accurate with weights and dimensions.
You can see the differences in the rate comparison table on this page: https://www.pirateship.com/usps/priority-mail
Look out for Dimensional Weight Pricing (DIM Weight)
If your package is big but light, it could get charged based on the dimensional weight instead of the actual weight. This applies if the package is larger than 1 cubic foot (for example, if the dimensions are 13x13x13") and being shipped to zones 5-9. To determine the “dimensional weight,” multiply Length * Width * Height and divide by the USPS volumetric divisor, 166. If the result is higher than the actual weight, the dimensional weight is what will be charged for.
To read more about Priority Mail weight-based, go to https://www.pirateship.com/usps/priority-mail-weight-based