Search this forum if you haven't already, lots of discussion on that here.
I'm pretty sure that you can do what you're looking for, but there may be limits. I looked into the type rating issue--but not combined with the ATP school, as I already have one-- myself last year.
Many schools accept the Chapter 33 (Post 9-11) G.I. Bill and have good experience dealing with the VA now that it's been around for awhile. Higher Power has it down to a science.
There's a $10,000 a year limit on type ratings, and it'll burn roughly the equivalent of a year of your benefit. You can't really get an expensive type (737 doable, G-V, not). You get no BAH while doing type training like you a do while getting a degree. Not really the most cost-effective use of the Chapter 33 benefit, probably more cost-effective if you're getting an ATP and a type together.
I believe there is also a limit on other (non-type rating) training though, 60% I think. Again, that wasn't my focus, so I'm not sure on that bit of it.
Not sure about what combining an ATP and a type will cost in terms of charge against your benefits, either.
I'd recommend a quick call to Sandy at Higher Power, she's super nice, super helpful, and can probably tell you what you want to know in 30 seconds.