Daedric Prince of Fate. Many seem to forget that aspect of Mora's control. However, even if this is the case, should you perfectly accept that you have to do what Mora wants in order to defeat Miraak, the only thing he asks of you, is perhaps a bit much, but nothing you are directly to blame for - he himself is. As for defeating Miraak in general though, this is due to the fact that the knowledge gained from Mora puts him a few steps above you, and without those same abilities(even if in a technical sense they are not directly required to face Miraak), you can't fight him at all. He isn't going to come to you, and Mora's reasoning for giving you the same shout(or the two words of it that you can't get elsewhere), is both due to the fact that he knows that Miraak seeks to escape the Daedra's control, and that he wants to see if you can actually face him yourself, i.e if you are worthy of taking the same place. Doesn't entirely matter if you want to or not.
Beyond this, however, it is not entirely clear at all what exactly happens to the Last Dragonborn after this. At least not yet, there is no official "This is what happened as a result..." statement or whatever else from Bethesda, so if you want to play it like you are lying to Mora completely, then it still works. If only because unlike Miraak, you never become beholden to Mora's power.
The worst thing Mora does is kill Storn, and that is something the man wasn't entirely willing to accept, but still did so. The way I see it, the only problem I had at first with the questline is the fact that Storn dies regardless, but notice that none if the Skaal, think that you are at fault for Storn's choice - HE wanted to speak with Mora after all, knowing that it was an inherently dangerous thing for him to do. Freya's "What have you done?" is NOT directed at the PC, but Storn himself. Everything aside, Mora's bidding or not, you are still doing it to face Miraak, who has done things that your own character is unlikely to do. It likely isn't Mora that wanted Miraak to take control of Solsthiem, but something he was able to do because of what he learned from Mora.
The Dragonborn DLC is not about choosing a side like Dawnguard is. Even with Dawnguard the choice is still flimsy, but just those it needs to work. But personally, I've only ever played one side, if only because I cannot stand how Isran becomes later in the questline.