I have no doubt that if you take a very agressive route you can be a very capable diver by October, as long as you catch on and learn fast.
Just FYI, my wife and I started diving last Jan. and we already are working on your Dive Master Cert. we have become very capable divers in a short time, but we dive a LOT. Most people maybe get in 10 dives a year. We are going to have over 100 dives in the last year by time April comes around. Along with those 100 dives they span the board on types. We have dove in pools, quarrys, Epcot living seas (that was fun), Niagara River dirfts, Lake Erie, Cold Ocean, Warm Ocean, Shore, little boats, huge boats, deep wrecks, shallow reefs, you name it. We even did some basic cavren diving with an instructor in FL.
So what you want to do is possible for sure.
As for your equipment, to start our shop will not even think of having a new student in a BP and wings. We have one of the best tech instructors in the area up here. He owns a BC and a BP. When he does pool classes and simple dives he wears his BC most of the time. He wears his BP on deep dives and tech dives.
So in the end if you start doing the heavy stuff yes your going to need a BP, but thats a ways off, yes maybe 3 years or so. But for the time being and when you travel to tropical destinations you will want a regular BC to bring. You typicaly own a small claw hammer to handle most jobs around the house but you also may have a sledge hammer to bust concrete if you need to. Same thing here.
In your basics class you will be doing gear removal skills. A BP by design does not have any quick release clips and only 1 buckle which is a threaded type sometimes. They are a little hard to get on and off. A wing only gives back inflation (and some BC's) when on the surface this will tend to put your face in the water. A BP does not have any pockets, most tech divers always wear a dry suit and use those pockets. Weight is also an issue, BP and wings weigh more than a BC when you travel.
There are many reasons you are going to want both if you get to a point where you need a backplate. It all comes down to the right tool for the right job. Don't think of a BC as entree level, I paid like $500 for my BC and its a very nice BC and serves me very well, same for my wife.
Also when you do your basics class don't buy gear right away. A shop will normally have a couple different types of rental gear, ask to try different stuff in the pool and your open water dives. This way you can kind of see what you like and don't like.