You can use bigger tires on a MGB. I went to lower profile and wider tires. I have 195-60-14 but you will have less ground clearance problems if you go to 175-70r14's which is what I used to run. And they are available lots of places.
If you want to keep it period looking, there's the option of 165-15, used on the GT. I use 186/65-15 because I have hubs and wires. Any wide would rub my wheel arches.