The pole thing definitely wouldn't fly with the spirit, and it's not good to have a bodyguard that feels you cheated it.
Guys look I don't need help getting around the problem, I need to do it the hard way. I need help getting through the problem.
So the way I'm looking at it, "the polar solution" is not, in fact, "getting around the problem." Rather the reverse -- it's HARDER. Consider that "carry a flame around the world" is a normal(ish) thing, happens every 4 years for the Olympics (every 2, I guess, since they do summer/winter Oly's on an alternating-biannual basis). So, it's a fundamentally "expected" thing, and it can be "known" without getting any special attention (and generally, "attention" = unwanted, for wizards in the Dresdenverse). Sure, you need to obscure (from the Mundanes) the fact that you're doing it magically... but hey, do-able and not even very challenging, really.
But take a FLAME up to the North (or down to the South) Pole? Carry it in a ginormous frigging CIRCLE around the "pole"? Coldest places in the WORLD, them. Often windy. Fundamentally as flame-hostile as anything outside the water can get, really; and you also need to make it "known" in some sort of symbolic magical way...
You just performed a magic circle of flame around one of the two most-symbolic "cold spots" in the ENTIRE WORLD.
From the "flame spirit" POV? I think it'd look like a "flame rules, cold drools!" move (possibly even annoying Mab). Serious, SERIOUS Brownie Points with Fire spirits, for whoever pulls THAT stunt in a manner accruing fiery/magical oomph!
And... um... you want to do it "the hard way," huh? Really... I don't see it getting a lot harder than that...