I did the HPDE ladder, got my comp license, raced W2W and now instruct.
For the goal of being in TT I think HPDE is the way to go. The typical way I see things progress for people is:
Group 1 (beginner) -- two weekends. Instructor assigned.
Group 2 -- probably 4-6 weekends. Instructor optional.
Group 3 -- duration is your comfort level. Check ride from instructor.
Group 4/TT/Open passing with no point-by -- you'd better be ready.
The most common error people make is wanting to get out of Group 1 or 2 too early. Gr 1 can be frustrating as there are a lot of beginners, many with less than stellar talent that need lots of coaching. But, duration in this group is generally short. Depending on the region Gr 2 may or may not assign instructors. I would request one, at least for a couple of sessions. Each instructor will bring something new to your table and pick out some of your negative quirks. Some instructors are more skilled than others. I've had--and been--an instructor that rode with someone and said "I can't teach you anything". Some drivers have risen to my skill level or possibly beyond. At that point you need to look toward more professional instruction, probably from competition drivers who are winning (I aren't one). The purpose of HPDE instruction is to teach you the line, proper track manners, get you comfortable running in traffic and be comfortable and safe at speed. Yes, we do strive to improve your driving but that is up to you to work on at a certain point. Driving in the open passing group with instructors and TT guys is not for a guy with two weekends under his belt. Scary for you and dangerous for the rest of us.
A comp school is designed with one thing in mind: get you a comp license. W2W is a different world entirely and while I don't doubt you will learn quite a bit from a GOOD comp school you will gain so much more from several weekends of HPDE with several unique instructors. I'm very comfortable on track with 90% of the guys I run with in Gr 4 and the ones I'm not are easy to spot...they usually drive really expensive cars.