For the majority of cases it has no affect on V1 primarily because the V1 is based on a fixed CG position, the only time that you will see a change is if you use an Alternate Forward CG AFM appendix.
Hi Mutt. It depends if your company permits the use of semi-permanent or 'fixed' de-rates, such as TO1/2/3 etc. for those that don't and use TO or D-TO you are correct. Some airlines believe it to be to 'risky' to use fixed derates.
As to the CG and improved climb. You are right. It's been around a long time. these days lo-co EU operators look to gain every single perf advantage and bundle them together for a higher PLTOM or full flex and min flap. Hence why I answered in that way. But yes. You can certainly do an improved climb with a regular, or even fully forward CG. You would however burn more fuel. Different operators, types, same runways, different ideas.
As we don't know the type or where they are sat in the OP's case, this is intended to be a general sweeping comment on the way some, not all, lo-cos go about arranging their aircraft for a cost saving takeoff. If your company uses CARD or ACARS for perf calcs, try using fixed derates on different intersections to see the whole range of possible V1s. A 'mean' V1 is usually settled on by the system, depending on whether it's octopus,CARD, navtech, EPOC etc etc