Will I fail a skills test for doing a constant-aspect climb-out?
I fly in the UK, and while there is certainly a "standard circuit" (ie pattern) in the syllabus there are many, many fields that have a non-standard circuit for one reason or another. This is often to avoid antagonizing unfriendly neighbors. 

There's also the fact that the NAP is not a rule or regulation, it's more how the airfield management would like you to fly. 

I personally don't like the constant aspect climb-out idea at all, it will mean you are paying more attention to aspect than airspeed and looking out for other traffic. It also means that the pattern traffic is not predictable for other visiting your airfield or by others in the pattern. So I would push back on your instructor on this. 

As for whether you will fail your skills test if you do it would depend on the examiner's view, and what they are looking for. While a constant aspect departure shows skill, I personally would take a dim view of people doing it despite the published departure procedure, even if it is non-binding. I would discuss it with your examiner ahead of time, and practice your circuits how they say they want it, or at least discuss it with them on the day.