Summary: eat the rabbit. Every single bit of it.
With 1 rabbit per day, you are in starvation while probably not exceeding your normal capacity for daily protein digestion, though it may very well exceed the protein digestion capacity if you are in total starvation (also without protein) for a prolonged amount of time.
In any case, you should stretch it out over a longer time.
To add to @bob1's answer, according to the German wiki page on the topic, there are two known main contributing factors to rabbit starvation:
- Too much protein together with
- insufficient total energy intake.
The "untrained" (wrt. protein intake) human body can digest up to ≈ 200 - 300 g protein per day, that's about 3500 - 5000 kJ. For comparison, a medium active standard male (1 h/day medium heavy work - which doesn't seem much in a survival in Russian winter scenario) needs 12500 kJ/day corresponding to the meat of about 8 European rabbits.
In turn, this means that no more than 30 - 40 % of the needed energy can possibly come from protein.
Plus (or rather, minus) for long term life as a hunter as opposed to survival for a few weeks some of that protein to be used for protein (muscles, connective tissue, bones etc.): maybe some 80 g (rough guesstimate based on numbers from Egan, British Nutrition FoundationNutrition Bulletin,41, 202–213, 2016.)
That would amount to pure protein diet leading to something in the order of magnitude of 75 - 80 % caloric intake deficit. In other words, severe starvation.
In addition it seems that unlike total starvation, in rabbit starvation the hunger feeling (craving for fat and carbohydrates) never ceases.
Note that also the English wiki page linked in the question cites a description that the sickness sets in after about a week of eating as much rabbit as possible but basically nothing else. I.e., considerably more than just one rabbit.
The survival situation.
Meat of a wild prey animals is very lean, often around 3,5 % and 20 - 22 % protein. Domestic animals like domestic rabbits or cow have roughly twice as much fat (7ish %) according to this website comparing venison with the meat of corresponding domestic animals (sorry, in German but the numbers are internationally readable) (at least when grown up, growing animal meat like veal can be very lean).
Your caged rabbit would already be better in that respect than a wild hare.
Also, as @bob1 pointed out, there are more fatty parts of the rabbit body: bone marrow, spinal marrow, brain, inner organs such as liver and kidneys (though the latter may not contain that much fat in lean rabbits - but I didn't find numbers). If the cage rabbit was someone's bunny always living in a cage, chances are that it has some more fat deposits. You'd clearly want to eat them.
Last but not least, let's assume the rabbit weighs 2 kg*. The meaty parts still with bones will be 50 - 60 %, sorry, German again, so 1 - 1.2 kg plus maybe 100 - 150 g of internal organs for eating (Brown et al, ORGAN WEIGHTS OF NORMAL RABBITS, 1925.). Assuming 20 % protein i.e. lean muscle tissue, ignoring bones etc. that's about 220 - 250 g pure protein (3700 - 4250 kJ) and maybe 10 - 20 g of fat (400 - 800 kJ).
So eating the whole rabbit in one day would get you just to the upper limit of possible daily protein intake.
For practical reasons you'd probably want to stretch it out a bit, so you still have something substantial to chew for the next days - but the situation will overall be just normal starvation, no or only small protein excess.
Longer term, you should limit yourself to not more than 1 rabbit per day to avoid the protein excess.
In order not to loose anything, salvage or produce a pot from the remainders of the plane and make a stew of the thing.
In addition, as this is about surving, I'd for sure include the rabbit blood and I'd seriously consider whether the stomach to cecum contents and cecotropes would be considered veggies under these conditions. Droppings and urine not, obviously, and if there's sufficient (liquid) water, the kidney will do with some watering.
* For European wild rabbits 2 kg is quite a big animal. Domestic rabbit breeds that are bred for meat production slaughtered at 2 - 3.5 kg (but can get much bigger when fully grown), pet breeds are usually smaller.
Edit: I forgot that the rabbit magically appears after you are already 2 weeks in total starvation in (Russian) winter. In that situation, it's probably a last and pleasant hallucination.
Anyways, magically, you are capable and the rabbit is real. Then you'll need to start eating slowly to avoid refeeding syndrome: even if there are almost no carbohydrates in the rabbit stew, and the whole thing in terms of energy intake is not even 2 days of starting recommendation for refeeding, my guess is that your liver has been economizing for a while already, so in that situation the possible daily protein intake is lowered). Also, the electrolytes are not in the required balance, comparably too few K⁺ in relation to the Na⁺ - stomach and gut contents ("veggies") may help a bit in that respect.
So maybe only a bit of the broth for the first meal, and from then on in small amounts to strech out that stew over several days.