Shipping with the Square Store is very basic as you know and doesn't work for all businesses.  If for example don't charge shipping because I can put it in an envelope.  That doesn't work for everyone.

Here are a couple of options

Create a flat rate that you know you can get most of your products out in.  USPS flat rate box is $X and I can get X in it.  Then if larger items require a larger box you adjust those items only.

Check out addon app Shipstation.

Upgrade to an e-commerce option that fits your needs better.