It sounds like a total screwed up mess.
You stated the applicable welding standard is AWS D1.1. You didn't specify a year. Depending on the year, the prequalified WPS must use a CV power supply. If the edition imposed by the customer is prior to 2008 (or so) CV was not a prerequisite for prequalification.
Assuming your client has imposed an earlier edition, CV or CC is not a prerequisite for prequalification. The welder is qualified using a WPS that could be prequalified or one that has been qualified by testing, or possibly a SWPS purchased from AWS. Assuming the welder takes the performance test with a prequalified WPS, the welder is qualified for CV/CC and EP/EN because those variables are not essential variables for the purpose of the welder's qualification.
If the customer has impose a more recent edition of AWS D1.1, the issue of CV or CC only comes into play if the contractor is writing the WPS as prequalified. If that is the case, the WPS is not prequalified and it is in error and should be considered to be invalid by the Engineer or Owner.. The WPS for FCAW/GMAW must be qualified by testing per clause 4. If the WPS is qualified by testing, the PQR must indicate the type of power supply and the polarity used to weld the test assembly.
It could be case where the person recording the test data on the PQR made a mistake. Just as plausible, the individual writing the prequalified WPS made a mistake and listed the power supply as constant current. It could be a case where no one wants to admit making the mistake and no one caught the problem during the review and approval process.
Keep in mind that AWS D1.1 allows the Engineer to circumvent, modify, add to, or delete the requirements of AWS D1.1. Such changes are to be addressed in the project specification. When there is a conflict or a question of the applicability of a clause, it is the Engineer's prerogative to issue a decision. The Engineer's decision stands unless a technical inquiry is made to the D1 committee and an interpretation is rendered.
If your customer, representing the Owner, is demanding that the welding be performed in a certain way, as long as the demand is in writing, that is the way the welding should be done. If there are problems down the way, whether it is nonconforming defects, failure to perform as expected, etc. your employer could face an uphill battle. If your employer tells you to do what the customer calls for, you do it and document every conversation and question asked about the customer's demands. It will come in very handy if there is a failure traced back to the welding, you can expect to be on the witness stand explaining why the welding was done a certain way. Trust me, at that point everyone from the Owner, contractors, mangers, supervisors, and the foreman will develop amnesia. The documentation you have to support your actions and activities will be your savior or the lack of documentation with be the rope around your neck.
Good luck my friend.
Al