For this: "the total vendor payments paid to support that client"

One every purchase transaction, did you list your Two Sided Service item, allocate the Split amounts per Client being served? That would also be the point at which you get to mark it Billable. Example:

I pay a hosting provider for hosting 4 (3 are clients') websites. I list this on the Purchase:

Hosting package $100 Smith Billable

Hosting package $100 Jones Billable

Hosting package Unlimited Plan $500 Doe Billable

For Expense, not Item: Hosting Package (my own site) $50 not Job tracked

= $750 each purchase I enter

The Service item it for Customers is linked to COGS and to Revenue.

Now I invoice you for your billable item.

Now I have Customer Profit, Item Profit, and P&L by Customer has the Tracked Cost item flowing to COGS.