I thought about this but haven't tried it yet:
Ideally use a head-shot deer.
Instead of bleeding in the usual manner, dissect out the oesophagus as normal but use the same cut to find the jugular vein (internal jugular and carotid artery together, I think).
Follow it toward the chest, trying to free it up along the way.
This should give you a length of vessel long enough to get into the neck of a clean receptacle.
Ensure the carcass is either hanging or lying in a head-down inclination.
Cut the exposed vessel at the head end and tuck the cut end (the side that is heading to the chest) into your receptacle.
Marvel at huge volume of liquid black pud draining into receptacle.
if no flow, look and feel for a valve in the vein, cut the vessel again on the downstream (heart-side) of the vein.
Let me know how you get on!