(Hi Cyril, welcome to Physics.SE!) You already received a couple good answers here but these mostly deal with Newtonian physics and don't address your actual question/claim, namely that special relativity should predict the bending of light. (Which is why one could indeed say that the other answers actually answer an "edited question", as Dale puts it in his comment to your question.) So I think it is in order to also point out to you why your arguments involving special relativity (SR) don't work and SR indeed does not predict the bending of light.

according to $E=mc^2$ [photons] also have a tiny, but non-zero mass equivalence

Photons do not have a rest mass and the energy-momentum relation $E=mc^2$ you cited holds only for massive particles at rest. In particular, it doesn't hold for photons. The correct and full energy-momentum relation (for any particle in any frame) reads $E^2=(mc^2)^2 + p^2c^2$, where $p$ is the momentum and $m = 0$ for a photon (and any other massless particle).

this should apply to a hypothetical photon at rest, not to real photons at velocity c

In SR there is no frame in which a photon would be at rest. This is because an object moving at the speed of light in one frame will move at the same velocity in all other frames. In order to understand this a bit better, I recommend you take a look at the book "Relativity, Groups, Particles" by Sexl & Urbantke which presents a particular nice derivation of the laws of special relativity (i.e. the Lorentz transformations) from a small number of reasonable assumptions (transformations between inertial frames be linear et cetera). Starting from these assumptions they arrive at a point in their derivation where they end up with a free parameter (I think they called it $k$) which they can choose at will. It turns out that if this parameter $k$ were chosen to be zero, Galilean mechanics would follow. A non-zero $k$, however, would imply that there is a "universal" velocity $c$, related to $k$ via $c^2 := 1/k^2$ (as far as a I remember), which is universal in the sense that any object traveling with velocity $c$ in one (inertial) reference frame would appear to be moving with $c$ in all other (inertial) reference frames, as well. Needless to say, experiments dictate that $k$ should indeed be non-zero, as we observe light to travel at the same velocity in every reference frame. The result of Sexl & Urbantke's derivation for non-zero $k$ are then the famous Lorentz transformations of special relativity. It turns out that these transformations leave the so-called Minkowski metric and, in particular, every particle's 4-velocity and 4-momentum invariant. It is from these structures that all other relations (like the energy-momentum relation and the invariance of mass$^†$) follow. In particular, it follows that photons cannot have a mass.

To summarize: Once you accept the fact that there 1) is a universal velocity $c$ which is the same in all reference frames and 2) that photons move at this universal velocity $c$, you are immediately forced to conclude that photons cannot have a non-zero mass and that there cannot be a reference frame in which they are at rest. So your argument that they should be subject to gravitation in a theory of special relativity (subject to Newtonian gravity) does not work.

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†) Here I assume that "mass" is defined as the Minkowski norm of the particle's 4-momentum $p$, $m^2 := -p_\mu p^\mu$. The fact that this definition makes sense again follows from comparing predictions with experimental observations.