Transfer pricing and VAT

The selling company is registered for VAT, so sales of VATable items to any buyer should be made including VAT.

This looks a little like an accountancy teachers' staff room question rather than one that would crop up very often.
I was trying to think of a company that might have a manufacturing subsidiary which was selling above the VAT threshold which I can see is very realistic.
Then I thought about company B. This might be a newish retail subsidiary which isn't turning over enough for VAT registration, or might even be a recent acquisition.

Look at the part of company A where the goods are made that are sold to company B. If the raw materials bought are not subject to VAT, can the raw materials be invoiced to company B directly by the supplier? This would then reduce the invoice from company A to company B to just the conversion costs.

Very often agencies in the service sector will only charge their fees to the client, but expect the client to pay the suppliers direct. I recall a DJ booking agency which did exactly that.