Much depends on how you are seasoning your braise to begin with. You could do it cacciatora style---in which case mostarda di cremona would make a nice table condiment. Thumper could also be served over paella or alongside a ratattouille.
Starch or veg? For starch, w/ all braises I particularly like soft creamy polenta. I make mine w/ cream and stock, and a lot of parm at the end. Mushrooms would add a lot to it as well. Chanterelles, shiitakes..... An accompanying green veg should be simple and clean tasting because of the richness of the braise and the polenta.
Broccoli, green beans,sauteed greens....
Even wild rabbit is fairly mild and therefore gives you a broad palette to work with. I would suggest you first think of the local produce that is peaking near you, choose what you like and then search here for good recipes. For instance in much of USA right now we have our best tomatoes, sweet corn, eggplant and many stone fruits (peaches, plums, nectarines) are peaking. I notice you're from Australia so I have no idea what produce is peaking near you.
I have found that braised leeks go really well--just wash them, cut in chunks, and put them in a dish with a little chicken broth, covered, in the 400 degree oven for 20-40 minutes (depending on how soft you want them). Also, the same with whole carrots and red potatoes with whole mushrooms. My hubby is Peruvian and likes a quinoa salad cooked with chicken broth and then mixed with fresh chopped cilantro.
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Broccoli, green beans,sauteed greens....