Sad that you're not allowed carbs.... One of my fave things to do with backstrap is to mince it up really fine and either stuff it into momos (Tibetan dumplings) or else make Stroganstuff (heavy on the fats and pastas.) But here goes! I do have a lovely Syrian recipe that I'd normally use lamb or goat in, but which should work extremely well for venison.
Begin by roasting some tomatoes and eggplant on the grill, until the skins are good and charred and the insides pleasantly mooshy. Cube those insides up, and add in generous (and I mean really generous) amounts of fresh parsely, along with a handfull or so of chopped raw Spanish (ie red) onion. Salt to taste.
Meanwhile, marinate that venison in a mixture of beer (red ale preferred, but lagers will work), tomato juice, and lime, with tons of crushed cumin seed and mint. That goes on a screaming hot grill until the outsides are just seared, and then it's sliced over the tomato/eggplant stuff. Toss on some walnut chunks.
Serve with thick sour yogurt. It's teh yum.
EDIT - I just re-read your intial post. Tomatoes are a no-no? That means eggplants and other solanums are out too. Boo-urns. Then maybe try this:
Flash-grilled venison, rubbed in cajun blackening spices (a mixture of chili/cayenne powder, milled oregano, milled thyme, garlic powder, black and white peppers, paprika, a bit of salt) sliced over lightly grilled Belgian endive halves, and the whole schmoo drizzled in olive oil. Sides: lightly steamed asparagus smothered in mango chutney, mesclun salad dressed with simple oil and vinegar.
This is challenging, you know. My approach to the salt kitchen is much more fat and carb oriented, due more than a little to my altitude (where fats and carbs aren't just good, but essential to my survival).
Here's a question: can you get your hands on Lupini beans, preferably fresh but if not, canned? If you can, they're a complete protein and quite low on the starches end of the scale, and there are some intensely good sides to be made of them.