If the customer specifies AWS D1.6, one must comply unless an exception is taken when the quote is submitted to the customer. Once the contract is signed and the PO accepted, the contractor is obligated to meet the terms and conditions of the contract.
If the contractor doesn't have the appropriate standards referenced by the contract documents, it is incumbent upon the contractor to obtain them. Compliance to the referenced documents isn't usually an option, but a requirement.
The customer can claim breach of contract if the contractor doesn't meet all the conditions of the referenced standard. In many cases, the customer can seek redress and damages from the contractor for failing to meet the requirements of the contract.
There are differences between D1.1 and D1.6, if there were not, there would be no need for D1.6.
Best regards - Al