Thanks for giving us something to do on a relatively boring Saturday evening shift. There are different views on this, but it has always been my belief that coordination would be required by Departure North with BVI tower to turn an aircraft within the Class D airspace, since this is not addressed in the PIT/BVI Letter of Agreement.
I think the debate begins because we separate airplanes and not airspace in the real world. I'm going to guess and say that you are attending CCBC for ATC. They will teach you that the 7110.65 says this...
2-1-14. COORDINATE USE OF AIRSPACE
a. Ensure that the necessary coordination has been accomplished before you allow an aircraft under your control to enter another controller's area of jurisdiction.
b. Before you issue control instructions directly or relay through another source to an aircraft which is within another controller's area of jurisdiction that will change that aircraft's heading, route, speed, or altitude, ensure that coordination has been accomplished with each of the controllers listed below whose area of jurisdiction is affected by those instructions unless otherwise specified by a letter of agreement or a facility directive:
1. The controller within whose area of jurisdiction the control instructions will be issued.
2. The controller receiving the transfer of control.
3. Any intervening controller(s) through whose area of jurisdiction the aircraft will pass.
That's all good in the academic world. The real world is a bit different and you may see some corners cut. We do it many times a day, but will have to take responsibility if something goes wrong.
Your question also addresses altitude. We cannot stop an aircraft at an altitude lower than 3000 ft. That is the initial altitude assigned by BVI. Our minimum vectoring altitude is 3000 ft in that area. We are permitted to climb the aircraft and most likely will on initial contact.
If you are a CCBC student I would not recommend telling your Approach I or Approach II instructor that you can't turn the aircraft off BVI and AGC until they are clear of the Class D. I used to teach that class. The difference between a 230 heading and TVT is negligible. That's why you release the aircraft on a heading as close to on course as possible.
w0x0f