Why would I not have received orders yet for AIT & my next Duty Station (PCS), when my AIT report date is just under a month away?

You're reclassing, right?
You went to your Career Counselor in order to reclass and this is all part of your Reclass. Why haven't you gone to your Career Counselor?

Also, you're not TDY Enroute to Fort Lee, you're going to be TDY enroute to your follow on duty station with TDY at Ft Lee. This is important because TDY Enroute to Fort Lee means you have PCS orders to Fort Lee which will be confusing to anyone trying to help you. You are not currently TDY Enroute anywhere but you will attend in a TDY Enroute status.

HRC is not in Ft Lee, it is a separate command; HRC is located at Fort Knox, they are the people who place you on assignment. You probably mean the MPD at Fort Lee. MPD at Fort Lee can't help you and chances are anything they tell you will be wrong. They only deal with students after they arrive there and have no ability to get you your orders.

What was supposed to happen is that you should have been placed on assignment 4 months prior to your class start date. Since you were not placed on assignment your Career Counselor has to submit a request for HRC to place you on assignment. Then you receive a packet from levy section with a 5118 to fill out. You turn that in and get orders. Then you turn that in to get your DD1610 authorizing TDY so the Army will pay your TDY.

However, there is no chance in hell that you will get an assignment (that takes a week or two), turn in your levy, receive your orders, turn those in for TDY authorization, plus 10 days to clear and one more to travel, prior to your class report date. Your Career Counselor will need to renegotiate your contract for a later date. The next earliest class starts in July so you'll want to hurry to start that process. Also, your CC can't renegotiate a contract unless you are fully qualified to reenlist - for instance not flagged or barred. Since you are most likely single at your rank you will need to remember that you have to clear post prior to going to AIT, you will not be authorized to return to clear after

You need your contract and school date (ATRRS reservation) documents. You can do you 5118 with those as you have a hard slot. Once those are complete, you can submit them to levy and receive your physical orders

Have you received your ATRRS Enrollment Notification for your course?