Hmm, let's see.
In terms of strength, Aconologia wins no question. Large Country vs Large Island. However, beating Cole to death isn't really an option. He can easily put himself back together from getting ripped apart at the atomic level. So Acnologia can't use raw strength.
However, Acno has a few things going for him that may help out. He could definitely BFR Cole to his time dimension and leave him stranded there... problem is, since this is in character, Acnologia would most likely follow Cole to this dimension, an promptly be killed by Cole's matter absorbing black hole or a Ray Field blast.
Now magic absorption is also another factor to consider. Since verse equalization is in effect, I'm assuming lighting, fire and Ray Field radiation = magic. I can definitely see Acnologia simply eating Cole's lightning and fire based attacks, he did laugh Laxus's and Natsu's elemental based attacks.
However, and this might be an NLF, could Acnologia devour Cole's Ray Field Blasts? It's basically super radiation that kills anything without a Conduit gene. Even augmented humans like the Corrupted and Vermaak 88 couldn't survive said blasts. Even if you somehow survive the Blast, the Ray Field Plague is still something to consider. It was so deadly that it killed people the same day they started showing symptoms,
Even if Acnologia could safely devour the blasts, which is debatable, he still can't put Cole down or survive his matter absorbing black holes. Worse comes to worse, Cole'll just turn himself into the black hole, teleport next to Acnologia, and then it's game over.
Also, never forget Ionic Drain. All Cole has to do is activate it, and if Acno is in range, he's done for. He won't be able to escape because, 1) his body is having the life and electricity drained from it, basically shutting it down, and 2) Cole can manipulate gravity to create a vaccum to pull him in.
Also, Power Absorption. If Cole can touch Acnologia and keep his grip, Acno can say goodbye to several of his techniques. Maybe even more if Cole can keep a hold on him.
Tl;dr Cole via regen, matter absorption, Ionic Drain, and power absorption.