I have used 7:00×16 inner tubes for many years in the 7:00×17 tires on my ’35 prior to using the radial truck tubes I mentioned earlier. They worked fine.
When installing tubes put a liberal coating of soap stone powder or if you can’t find that talcum powder in the casing to cut down heat from friction.
Some tires have a wide bead probably because they are made with truck casings. This bead extends beyond the inner edge of the rim and in time will cut the inside of the inner tube causing failure. Check the width of the bead on your tires compared to the width of the step of the rim where the bead rests.