[Wrennie]

Is it possible to commission an independent chemist or an organization to synthesize a barbiturate for me that hasn’t been manufactured in over two years?

[MarkPawelek]

If this is drug meant for people - see your doctor for an alternative
If it's meant for animals - see a vet for an alternative.
This old barb you want will probably be a class C controlled drug in UK. Which means it cannot be legally made, traded, bought or sold without a prescription or government license.

Elsewhere, the precise laws will be more or less restrictive.

The easiest way to get a barb is through a doctor's prescription. It's not easy to get a license to make controlled drugs in UK, so only a person able to justify to cost, paperwork, and legal fees will do that. A person such as a large manufacturer like Sigma-Aldrich.