Can carnivores survive on a (processed) vegetarian diet? Why or why not?
With arbitrarily broad meaning of "processed", yes. If absolutely nothing else, elemental purification and inorganic synthesis of something chemically resembling meat in nutrition is possible. Prohibitively expensive and stupid, yes, but possible. So then the question becomes exactly how much processing you need to do. Without clear bounds on that question and inadequate research on the metabolism of obligate carnivores, nothing closer than a broad stab at an answer makes sense. That said, here's a broad stab.

For instance, you'd probably need to synthesize vitamin D3, carnosine, DHA, and a handful of other things from plant extract, which is possible but difficult. Carnivores often have carnosine synthases, for instance. Cats have two distinct carnosine synthases. In terms of dietary sufficiency, that may or may not be enough. To be sure, you'd have to break it down to amino acids and build it back up again, essentially. Obligate carnivores are likely to have a number of other metabolic needs that aren't met with, for instance, pulverized soybeans. You're going to need at least one vat for fermenting raw amino acids into carnosine, and probably another for vitamin D3. Seawater microalgae contain lots of DHA, so that is probably something to look into. You could oversupplement alpha-linolenic acid instead of supplementing DHA, which might or might not work. There's not a lot of study into carnivore metabolism, so prepare to make some discoveries the hard way. Genetic studies can only take you so far, so don't try this on your favorite pet first.

In short: yes, you can raise carnivores on a diet made from plants and plant material. Rhetorical you, as in it is possible. It's probably not something you can do at home, even if you're wealthy.

(I'm assuming "feeding the vegetable matter to chickens" falls outside the domain of "processed" but it would be much easier than large-scale amino acid purification and fermentation.)