One possible confusion: The Darmon-Merel paper is from 1997 (not 2007), after Wiles and Conrad-Diamond-Taylor, but before the proof of the full modularity theorem of Breuil-Conrad-Diamond-Taylor 2001.
The modularity theorems for elliptic curves (generally) go via a p-adic modularity lifting theorem for $p = 3$, because $\mathrm{GL}_2(\mathbf{F}_3)$ representations are related to tetrahedral and octahedral Artin representations. One key technical aspect of Wiles' arguments is that one really needs to control the image of the $p$-adic Galois representation locally at $p$. When the representation (say coming from an elliptic curve) has level prime to $p$, this is not so hard --- either the representation is ordinary, or it is "finite flat." The higher the power of $p$, the more complicated the integral $p$-adic Hodge theory becomes. One should think of the main technical innovation of Conrad-Diamond-Taylor as understanding the technical issues for representations which become finite flat over a tamely ramified extension, and Breuil-Conrad-Diamond-Taylor as pushing these methods further to some wildy ramified cases. (This is why the Breuil-Conrad-Diamond-Taylor paper is subtitled "wild $3$-adic exercises" or something like that.) With B-C-D-T, one can handle the case when $27$ divides the conductor. Note that, up to twist, this is basically the largest power of $3$ which can arise from an elliptic curve over $\mathbf{Q}$. It should be said that there are further issues related to $p$-adic deformation rings for "potentially Barsotti-Tate" representations which required significant new methods, introduced by Kisin. So B-C-D-T were "lucky" in some sense that the case $27\| N$ turned out OK by their methods. (to be precise, the corresponding local deformation rings were smooth.)
The short answer: there is no "conflict" with the modularity theorem, it's just that earlier results used $p$-adic methods for $p = 3$ and required that the conductor be not too divisible by $p$. After the proof of the full modularity theorem by Breuil-Conrad-Diamond-Taylor, there were no longer any issues with $E/\mathbf{Q}$ of conductor divisible by $27$.
The question about computing the conductor is really answered in the article itself (Proposition 1.1)
(There is at the time of this answer a comment by reuns which completely irrelevant to the actual question, btw).