I'm happy to hear you ignored your track day buddies. During track days most decent vintage guys on bikes half the size embarrass their modern counterparts in the cornering bits, but that's not worth discussing further. I'm glad people scratch their itch on the track, racing or otherwise, then ride sensibly on the street.
I don't want to discourage you but have to agree with TG that the 350 twin is undoubtedly more popular. These machines have developed a cult following and they have race classes everywhere. In our ass'n we had one 350F that was really well built and ridden by an awesome rider. It hasn't been out in a few years though. These are great little engines, like a sewing machine, but will be expensive to make competitive. Compression & tight squish for sure but the intakes are choked and valves are tiny. A good twin can make 50hp and are lighter with lots of available go fast parts like TG mentioned.
That said there's nothing saying you can't have fun. Stripping every gram of weight and focus on making it handle, add race rubber so it's easy & fun to ride fast first. Then if "it sticks" (the racing part that is) engine stuff can happen down the road.
The search function isn't working lately so it's a bit tough to find a lot of info on here lately.
Some fast guys will tell you there is no need for extra bracing, and it could well be. A lot of us do add some bracing, but maybe we're fooling ourselves! Certainly it can be easy to simply add weight and that is going backwards. I have a friend in the biz and here's what he did for me.
Forget the steering head section below, otherwise he prefers thin wall chromoly tubing and thin material forming "saddle tees" for bracing. On a single downtube frame, one cross member above the swingarm mount but as close to the pivot as possible, (I used an "X" pattern originally) and one between upper shock mounts. If you're using a race style seat there is no need for any loop deal, just tie the two sides together with a straight tube, keep it simple. Saddle tees stiffen areas as shown, and don't overdo those either. Don't weld the whole way around the original tubing for saddles but try to limit the weld to one side.
Keep in mind that adding a member across the frame near the swingarm pivot must not be done unless the engine is bolted in, plus the swingarm is in place and properly tensioned otherwise getting the arm in place after might be more difficult!