Here is the pertinent reg (Part 61.39)
(b) Notwithstanding the provisions of paragraphs (a)(1) and (2) of this section, an applicant for an airline transport pilot certificate or an additional rating to an airline transport certificate may take the practical test for that certificate or rating with an expired knowledge test report, provided that the applicant:
(1) Is employed as a flight crewmember by a certificate holder under part 121, 125, or 135 of this chapter at the time of the practical test and has satisfactorily accomplished that operator's approved—
(i) Pilot in command aircraft qualification training program that is appropriate to the certificate and rating sought; and
(ii) Qualification training requirements appropriate to the certificate and rating sought; or
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So, in your case, at the time you take the practical test (check ride) will you be in the employ of a 121 or 135 operator?
Will you have completed their training for PIC qualification in the aircraft in which you will take your practical?
If so, your written can be expired and that's OK. The way I read your situation is that you will not have completed any employers PIC program in the Hawker so you will need to retake for your ATP practical.