Energy-efficiency aside, what are the chemical constraints on CO₂ capture and methanation?
I'm only directly attacking your direct question, "what sort of catalysts might be used, and what are the performance / lifetime issues with them?" because the rest is ill-defined. I'll mention some electrochemical conversion aspects. I hope to illustrate why your question is problematic.

There are several different products one could imagine reducing carbon dioxide to:


methane
methanol
formaldehyde
carbon monoxide
formic acid
ethylene


Each will require different materials and conditions, but they all require energy for this to occur. Not to mention, these are all difficult reactions. Thermodynamically speaking carbon dioxide is at the bottom of this energy well and to get out of it requires spending energy.

One approach involves directly electrolyzing on a metal. Cu in aqueous solutions can give methane as a major product, but it also evolves hydrogen and many of the others I listed above. There are a couple of problems with this but the major problem is that it often requires large overpotentials at rates of interest for industrial purposes and this means more cost in terms of electrical input. Another major problem surrounding nearly all of these studies is the selectivity of a catalyst: in water at pH 7 evolving dihydrogen is thermodynamically more favorable than reducing carbon dioxide to everything except methane and methanol. Unfortunately these two are the most complex, as far as chemical transformations are concerned, to make. So the other products are much more likely to be produced along the way. This is why people have focused on making simpler things such as carbon monoxide or formic acid.

Where does this energy come from? Right now coal. So we would end up creating more carbon dioxide than we reduced to other species. It reminds me of how diesel-electric transmission works in freight trains: a dirty engine turns an electric generator that generates electricity that is used on motors instead of the motor directly driving the wheels; I think we can call this diesel-electric transmission an indirect drive whereas the latter is a direct drive. It turns out that making the former is far easier in engineering rather than the latter. The same is true here. It's easier to generate electricity somewhere else then reduce CO2 rather than have energy directly fed to reduce carbon dioxide. You can get efficiencies better by going the latter, but engineering/technical complexities increase immensely.

As a means to address this electrical burden via the direct-drive analogy, people have looked at these artificial photosynthesis approaches, i.e. using light to  drive these reactions or at least help ease the energy burden. This requires photocatalysts and often with these come semiconductors. Some semiconductors are naturally good at reducing carbon dioxide, such as CdTe, but this better performance for generating methanol is in turn related to an actual corrosion process of the semiconductor. Semiconductors are very complicated and expensive to manufacture well. They tend to corrode easily when they work well with sunlight. They require catalysts and this introduces huge compatibility issues and manufacturing complexities. We'll just leave the indirect approach at the state  the field is, complicated and incomplete.

As for the direct approach:

it's complicated by the fact that carbon dioxide is fundamentally limited in solubility at room temperature. Hence why other people prefer non-water systems. These unfortunately add other complexities...

Actually while typing this I came to the conclusion that I could write a much more succinct response to you rather than continuing to babble on: The best candidate will involve a transition metal, there is no doubt about that. It may end up being a molecular species, but there are so many possible structures that it is hard to say if we won't find a simpler heterogenous catalyst that is more capable first. By Murphy's law, it will be the most expensive material imaginable. But right now, considering everything, you can always look at all extensive work that has been done on copper, but realize it isn't very efficient or selective.