First, great pick on the hotel. I love the St. George, and Via Giulia in general. I believe it's one of the longest contiguous streets in the historic center, and it's a really nice walk.

I haven't been to Rome in a while, longer than is probably reasonable to throw out restaurant recommendations since I haven't been back to see if they've held up. However, I'll give you a place nearby if you need something close - it's in the Michelin Guide still (not starred, but mentioned, which suggests its held up). Taverna Giulia - it's a guy from Liguria and they serve a lot of Ligurian dishes, so it's not where I'd send someone for one night in Rome, but in a pinch it should be a good meal with another region's dishes

Just so you don't think I'm nuts, I discovered this place about 5-6 years ago when we were hungry around lunch time and wandering back from the Vatican. It was a purely random wandering find, but when we went in, it smelled nice so we decided to stay.