Looking for an ecommerce platform that allows custom shipping options

Hi guys. I'm new here so sorry if I'm posting on the wrong section, or violating some forum rules.

I have a hand made ceramics store and I've recently been thinking of taking my store online.
I'm pretty much up to speed on most things, but there is one thing I absolutely need and can't seem to find on any platform:
I'd like to have custom shipping rates based on locations like neighborhoods, schools, local businesses etc..
Most platforms I've seen so far(Shopify, WooCommerce etc...) seem to only allow shipping zones based on ZIP codes or nations etc.
What I wanted was the ability to customize the checkout or even cart page of one of these platforms.
Since this is a local delivery business it is crucial that we should have drop down options of areas to choose from at checkout which would update the delivery price , instead of the regular method of entering an address zip code etc.
I was thinking there'd be a dropdown with a pickup option, and a delivery option. If they choose pickup, they'll choose a time frame.
If they choose the delivery option, they'll choose the type of location. For example if it's a school the customer shouldn't need to add address - just phone number & name - as I will have a list of local schools and they can pick which one they'd like.
The final price will be changed based on the dropdown values, and I would get the details of what the customers chose in the order details.
I know this won't all be possible without some custom code, and I'm willing to hire a programmer for it.
But for now, what platform do you guys think would be the best for all of this?
I'm open to any suggestions, even apps on Shopify.
Thank you for taking the time to respond.