Yes, since your benefits “began” in August (or worst case Sept), if you pass your wife will receive the benefit you are actually receiving. The changeover with her receiving your benefit should be effective the month you pass, since any benefit you are paid that month is for the prior month. Her lower benefit will stop effective the same month for which she collects your higher benefit. Therefore your waiting to age 70 to file will pay off for as long as either one of you is around. You should not have to talk to too many Reps until you find one that agrees.
This benefit of waiting until 70 has been publicized for many years as a survivor benefit for a surviving spouse who currently collects a lower benefit.