Having just now encountered this thread, and not having seem the other threads in which the OP is accused of having trolled, I may have a different perspective from the rest of the crowd. I also looked at his explanations and developed a theory of his issue, a theory based upon my own experiences. Perhaps I am wrong, but if I am not, then I believe I can give some meaningful advice.
From the comments, I sense the OP is sincere (not a troll), but is looking for justification for not taking the step to the BP/W immediately. I was (and in some ways still am) in that situation. A full set of cave diving equipment is extremely expensive, and if one lacks the financial resources to purchase everything at once, one looks for ways to to it in stages, buying only what is really needed at the moment. I myself have gone that same route. Even though I have what I need now for cave diving, I have not bothered to buy a singles wing and STA because on the occasions when I need to dive a single tank, I can always use my old back inflate BCD, or, as I did last week, borrow a singles wing and STA from a friend.
That leads me to my first suggestion.
The diver who loaned me the singles wing and STA is my tech diving instructor. I could not have gone down the tech diving path for reasons described above were it not for the fact that when he started his tech diving instruction, he reasoned that there would be people like me out there, and he accumulated an arsenal of equipment that students could use while they slowly built their own inventories. On my first dives with doubles, the only equipment of my own that I used was my dry suit and fins. I bought a little at a time over a two year period until I was pretty much all geared up myself.
This leads me to my second suggestion.
I agree with those who feel you should not do the singles cave instruction. I see it as a waste of your time. It really does not give you any valuable skills and will in fact delay your entry into real cave diving because you will be wasting your time better spent preparing for the real thing. You don't need to have your own doubles to have cave diving instruction. For my first classes, I flew to Florida, and I certainly did not fly with doubles. My instructor had what I needed, at no additional cost.
However, I do not advocate doing that now.
Instead, I advocate trying to find local tech instruction of the kind I described above. I bet you will find tech instructors like mine in your area if you look around. Build up your skills diving with doubles and a BP/W, even if it is borrowed. Slowly build up both a gear inventory and a skill inventory. There is no need to rush you cave training. You aren't going to get full cave anyway until you have really solid skills. Start honing those skills in the open water first.
Eventually you will develop the trim, buoyancy, and finning skills you need while wearing doubles. That will be the time to start looking for the right cave instructor.