Custom order list/composition

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I would like to create custom page in my template to achieve something similar to this mockup:

Problem overview.png

 I would like to have page when customer can create his own composition/custom order made from products added already to the store. Basically my products are made from sub-products and instead of creating lot of variants, I would like to give ability to create it by user. After choosing every product in a form, user will be able to click "Add to to cart" and all of them will be placed in cart (if there is ability to group them somehow in cart - it would be awwwwesome. But I understand that it can be tricky)

I don't want to use product variants because it's important for me to keep them separately in the store.

So basically I would like to create a form, where user can select products and add them all at once to the cart.
I also do not want to use any 3rd party apps.

All tips, tricks, guides and help is very welcomed.

Thank you!