If you are open to the idea of a larger, premium machine, then bikes like the TVS Ronin, Royal Enfield Hunter 350 and Honda CB350 will be the options we recommend. If you want to instead go for a commuter bike, there are loads of great options in the 160cc space, such as the Honda Unicorn / SP160, Bajaj Pulsar N160, TVS Apache RTR 160 4V and Suzuki Gixxer. 

Take a test ride of all the bikes you shortlist with a pillion to see what suits you best. 

Considering your height and your back issue, you will require something with good suspension, a low seat and a comfortable seating posture. Very few bikes amalgamate all three, so we have chosen to prioritize suspension over an ultra-low seat. Over time, you can work around that issue and even get comfortable with it. 

Some options in the 150-160cc space to look at are the Honda Unicorn / SP160, TVS Apache RTR 160 4V and the Bajaj Pulsar N160. All have seat heights under 800mm and have compliant suspension setups. 

If you want a larger machine under the 350cc bracket, then you can consider the Royal Enfield Hunter 350 as well as the Royal Enfield Meteor 350. The latter has a lower seat, but its feet-forward seating position won't be very comfortable over bad roads. The Hunter has supple suspension, and while it has a taller seat, its neutral seating position will be better over bumps. 

We would strongly recommend taking a test ride of whatever you decide to go with to make sure you're comfortable with the bike.