I thought of this too but the high throughput of fluid to get fast cooling would require a pressure so high it would destroy the organ. Running micro tubes through the capillaries would require significant advances in materials science.

But doesn't the loop-back happens in capillaries - which are biologically mediated?

Totally outside my area of expertise, but I wouldn't expect you can pump cold fluid into arteries and it'll just come gushing out the veins

What do you reckon, tamper with the existing fluid, maybe add some heat pipes into the existing large passageways? Not sure if they have the right shape for the kind of temperature change being discussed.