Under normal earth conditions, a human body only emits about 2/3 of its thermal energy as radiation. That means we dump a full 1/3 of our metabolic waste heat via convection + conduction.
Now, you said that vacuum is "a perfect insulator". In fact, it is no such thing. Its insulating effects are restricted to blocking convective and conductive heat transfers. It does absolutely nothing to stop radiative transfers, which is why good thermos bottles are mirrored on the inside.
Here, we have a problem. Because you said that the suit has "zero thermal insulation". But given that almost 1/3 of human heat loss occurs via the convection/conduction routes, and the suit maintains life support, I have to assume that the suit does not allow the astronaut to sweat away all their water to the vacuum of space. If the suit is indeed watertight, then it is insulating 1/3 of normal heat loss! So, the question, as stated, is not well-posed.
On earth, when humans need to operate in very cold environments, they usually layer clothes with different fabrics which serve different purposes. The innermost base layer is usually designed to wick water away from the wearer so that sweat can serve its normal function of removing heat from the body instead of building up on the skin.
There is also the ambiguity of the suit's material construction. To say that it has "zero thermal insulation" implies that it is a perfect thermal conductor. In that case, the suit should reach equilibrium with the wearer very quickly. Even so, unless the suit is also water-permeable, the suit will also be limited to the radiative throughput of the wearer's surface area.
Complicating the matter is the fact that a human is not a very good blackbody. A blackbody must be in internal thermal equilibrium. However, the body actively maintains more heat in the core than the periphery, and will shunt blood flow to the limbs in order to preserve core heat. There are thermal gradients all over the body, which is why medical thermometers are internal, and external readings are considerably lower (easily 4-5 C).
Finally, you have the fact that while the body can survive high heat-loss environments by shunting blood to the core, it has very little recourse for surviving pathologically low heat-loss environments. It cannot create a cold sink out of nowhere, in violation of thermodynamics (I mean, it could theoretically concentrate excess heat in a specialized organ for a finite time, but this just delays the inevitable).
So, while the lowest recorded body temperature is below 12 C, the highest is only 46 C. Even 41 C is associated with organ failure. So while the body can survive a loss of 25 C on the low end, on the high end, there is only about 4 C headroom for heat accumulation before you are looking at likely long-term injury.
Now, if you take away 1/3 of heat dissipation and force the body to dump that heat via radiation, you need to be able to do so without increasing core temp by more than 4 C. A blackbody certainly couldn't increase radiative output by 50% with just a 4 C increase in temperature. What your astronaut needs is a 50% increase in surface area: i.e., a decent-sized heatsink.