The answer is no. In what follows (see OP's comments below) we assume that the arrows of the category of lctvs are Michal-Bastiani smooth maps. Recall that a map $\Phi:E\rightarrow F$ from a lctvs $E$ into another lctvs $F$ is said to be Michal-Bastiani smooth if its directional (Gâteaux) derivatives of order $k$
$$ D^k\Phi[x](y_1,\ldots,y_k) = \left.\frac{\partial^k}{\partial\lambda_1\cdots\partial\lambda_k}\right|_{\lambda_1=\cdots=\lambda_k=0}\Phi\left(x+\sum^k_{j=1}\lambda_j y_j\right)$$
exist and the maps $D^k\Phi:E\times E^k\rightarrow F$ are (jointly) continuous for all $k\in\mathbb{N}$. This notion of smoothness is the one used in Milnor's treatment of infinite-dimensional Lie groups (J. Milnor, "Remarks on Infinite-Dimensional Lie Groups". In: B. DeWitt, R. Stora, eds., Les Houches Session XL, Relativity, Groups and Topology II (North-Holland, 1984), pp. 1007-1057) and Hamilton's exposé of the Nash-Moser inverse function theorem (R.S. Hamilton, "The Inverse Function Theorem of Nash and Moser". Bull. Amer. Math. Soc. 7 (1982) 65-222).
Michal-Bastiani smooth maps are also smooth in the sense of Kriegl and Michor (i.e. composing with a smooth curve leads to another smooth curve; some call the latter convenient smoothness), since the chain rule holds for Michal-Bastiani smooth maps. The converse is not necessarily true - both notions of smoothness coincide for Fréchet spaces, but in general they differ: conveniently smooth maps need not even be continuous, whereas Michal-Bastiani smooth maps are always so. A counter-example to continuity for conveniently smooth maps may be found in the paper of H. Glöckner, "Discontinuous Non-Linear Mappings on Locally Convex Direct Products". Publ. Math. Debrecen 68 (2006) 1-13, arXiv:math/0503387. In particular, if neither $E$ nor $F$ are Fréchet spaces, there may be maps between $E$ and $F$ which are smooth in the diffeological sense and need not even be continuous, let alone Michal-Bastiani smooth. Therefore, the natural inclusion functor of lctvs into diffeological spaces is not fully faithful. It is not clear from the above counterexample whether there are convenient diffeomorphisms which fail to be continuous or Michal-Bastiani smooth.