After starting to redo the analysis, I've come to realize this is a fairly complicated process, and instead I'll outline what I would do to solve this if I had more time.
First, some assumptions I'd make:
perfectly plastic behavior of the HDPE (i.e., the stress-strain curve is flat after yielding)
thermal convection from the flat surface is irrelevant (though it is taken into account for the pin)
the pin has an adiabatic tip
the metal is at a constant temperature
the pin pierces the surface when the energy absorbed by a cylindrical portion of the material directly beneath it is exceeds the toughness of the material multiplied by the volume of cylindrical portion of the material (I can explain this differently if desired.)
the contact resistance decreases as the HDPE melts (not sure how)
some material properties can come from this paper and others from this data sheet, and this book (this data may or may not be valid for the HDPE you have)
I'm not sure what value of the convective heat transfer coefficient for the pin is valid and would have to think more about this.
heat of fusion comes from this data sheet
Given all of this information, you can write 1D heat transfer solver for the Stefan problem to compute the temperature and thickness of the solid skin. From there, you can use the temperature of the skin to find the yield stress. My simple model suggests that $F = \sigma_\text{y} A_\text{contact}$.
Old answer:
The answer is likely zero force or nearly so, because the melting temperature of HDPE is about 266 F (it's reported a little higher here at 279 F), so the HDPE probably has already melted at the specified temperature. You'll meet the molten layer immediately upon touching the surface (at least if the temperature is uniform enough).
If the temperature range is wrong, I think I can develop an estimate assuming that HDPE is a perfectly plastic material, that the impacter is flat, and some other things, and then change my answer. Let me know if you'd like this.
Also, if you want an estimate of the force required to continue penetrating the medium, I can find an old terminal ballistics book I read that has some models for such things, but I don't know how accurate they are.