You need to understand that the lab techs and TP are talking Anaerobic or Lactate Threshold when the use the term “Threshold”. I know this for sure about TP and am pretty sure about this with your lab folks.
The information you want to find out as you enter this base period is the your AEROBIC threshold (AeT). Adding HIIT will move your AnT upward for sure. But if your aerobic base (AeT) is low you will never reach your peak of aerobic endurance as represented by AnT. When you have raised your AeT to within 10% (in terms of HR) of your Anaerobic Threshold (AnT) then for sure adding high intensity will greatly affect the AnT.
Build the aerobic base first and you will see much greater and longer lasting returns rather than going immediately to the HIIT prescription.