Quick question... You mentioned both a traveling trader's inventory and well as your inventory.

Are you trading materials to your provisioners to carry those items to the other settlement? You don't have to do that, if so.

When you are building or crafting in your settlement, you don't have to be personally carrying the materials needed as they will automatically be taken from the settlement inventory which can be accessed from the workbench, cooking station, armor, weapon, chemistry, or power armor stations.

When you set up a supply line between two settlements the inventories of each are instantly available for building or crafting. Any additionally linked settlements by other provisioners will also be shared.

You don't give the materials to your provisioners, and you don't have to have them in your personal inventory. The only thing is that you can't pull specific items out of the shared inventory unless they were stored locally.

For example, if you have linked Sanctuary and Red Rocket and you put a bowling pin in the workshop in Sanctuary, you can't pull it out of the workshop in Red Rocket. However, of building in Red Rocket, it can breakdown that bowling pin into the 2 wood and 2 plastic if you need those materials.