For instance, if I were to describe someone as being *near-*carnivorous, I'm have the goal of depicting them as being a heavy meat eater that includes very few forms of non-meat based food in their diet as opposed to being one who exclusively consumes meat.
I'm very much in favour of quasi- for this purpose, but near serves the purpose, as does mostly.
Quasi- used to show that something is almost, but not completely, the thing described http://dictionary.cambridge.org/dictionary/british/quasi?q=quasi
The prefix peri- may suit
enclosing, encircling, or around ⇒ pericardium, pericarp, perigon
near or adjacent ⇒ perihelion
"semi: a Latin prefix to a verb, noun, or adjective meaning 'half'".
Even though it means half, it is often used in context to mean "almost", especially in cases when it is applied to non-measurable concepts. Some examples are "semi-automatic firearm", "semi-parasitic creature", or "semi-arid climate".
I think that "pseudo-" could work in the context of "near carnivorous". Pseudo means "false" or "mock", so it would not work in all cases as "near". But "pseudo-carnivorous" for some reason sounds more natural to me than "semi-carnivorous".
I think para- comes closest, when used in the sense of resembling or similar, as in para-military, para-medic or para-legal.
Others you might consider in particular contexts are quasi- and crypto-.
According to Wikipedia, English has borrowed presque-isle from the French, to mean a peninsula with an unusually narrow neck. However, this seems to be the sole use of this prefix in English - so far.
Carnivorous is essentially a non-gradable (because it classifies) adjective. To get round this fact to describe an animal eating say 98% meat, one could use the construction 'almost exclusively carnivorous'.
Why not pen-, as in penultimate or peninsula?