Speaking from personal experience, having to do CAP or DP 1.1 (the new "phase III") as a starting point is considered on a case by case basis. For me, (now, this was 9 years ago), I was UTPNCM when I was a MCpl, complete with my small arms' qualification (now known as "DP 3A"). I had to do BOTC parts I and II (Chilliwack for both), Phase II Infantry ("CAP"), Phase III Infantry ("DP 1.1") and Phase IV Infantry ("DP 1.2"). Prior to attending Phase II Infantry, I applied to have it waived and attend Phase III Infantry instead. My request was denied. In my opinion, though I learned from my Phase II Infantry, it was a waste of my time and the Infantry School's time. Example:
As with most courses, the candidates are shown "the standard" in various PCs. For the section attack demo, Demo Platoon was short one person. So, the options for staff were (a) Do a section attack short one C9 LMG or (b) do a section attack demo complete with one cadidate acting as number 2 LMG. Guess who that guy was? (Hint: I see him every morning when I shave and brush my teeth).
So, it comes down to case by case. I've seen former warrant officers having to complete DP 1.2, and other getting it waived. The qualifications are a factor to be considered by "them", but it's not the only factor.