within walking distance, for food and wine (not HAUTE cuisine but more brasserie comfort food) I like Brasserie Lip.
There's a great small Spanish place on one of the side streets between where you asked about and the Hotel du Rhone; ask around, most should know it. Very cosy, decent selection of reasonably priced wines...
If you are willing to walk up to Old Town, 'Les Armures' Restaurant has a decent selection of Swiss wines and you can get venison and a half decent raclette.
Go up towards the Cornavin train station and there are two okay (but for me, sentimental favorites) restaurants amid all the tourist traps - Le Entrecote; and a Italian place.
Of course, you can try the all you can eat peal and eat shrimp (La Sirena?) on the wrong side of the Cornavin Station (the TGV side) and a large mug of cold cerveza...
The restaurants to the north and east of the Pont de Montblanc are okay but grossly overpriced, imho...
But for convenience, stay within walking distance of your hotel, and stick to the above. The best and most "elegant" places are too far...
but you specifically asked about a wine bar, and for a true oenophile, I think most of the true wine bars in the area you asked about will either disappoint, or give you a coronary from sticker shock.
Enjoy!
TM
"A Purist does not believe in gilt by association. Respect, AND disdain, have to be earned..."