While TurbFD will be the best (or even Blender if you want to wrap your head around that – but it fits your budget) you could also do this in AE with particular and use a displacement to give you that “refractive” effect. Its not actually refraction in the real world, its that the text is constantly changing depth as it rides over the changing geometry of the smoke. Add to that the depth that it penetrates: you can achieve this with multiple versions of your text element set back slightly and with different displaces.
Brian is bang on with the advice for lights and smoke in C4D but I worry that C4Ds volumetrics will be too large scale and fluffy for the kind of smoke “screen” you want. Pyrocluster tends to be best for volcano type eruptions (hence the name) but doesn’t do gauzy smoky stuff so well (as any volumetric puff will have trouble with).
Even with Turb FD for this kind of thing we usually have to add in Krakatoa to get the number of particle instances up into the billions to smooth out the smoke. But that would really kill your budget.
What about filming it? Dry ice in a trough of water works great!