When is one allowed to cook in a utensil that was designated for the other type of food (milk vs. meat)?

In connection with the previous question: does the fact that I don’t have another griddle pan put me in the category of “doesn’t have another utensil”? (I could fry in a regular frying pan or in an oven; however, the resulting taste and quality of food won’t be what I want.)

Since this utensil has a special use, and you don’t have another utensil like that, it’s considered as if you didn’t have another utensil.