First, I agree with other posters that, as is often stated here, an invitation is not a summons, and you may decline for whatever reason you may wish/need to do so.
As for milestone birthdays, I just turned 50, and while I, too, wanted very much to celebrate in a way which felt significant to me though I don't feel at all strongly regarding my birthday other years, this one simply was not just an objective milestone, but one I truly felt was important *to me* (in a good way, I will add).
That said, and while I wish OP the best of celebrations for her milestone, I don't see it as the "key" reason to not attend her sister's wedding. If, as seems to be so, OP has seen her brother once or twice in 13 years, and the rest of her family once in that span, then being present for family events doesn't seem to be an established practice (surely there have been other events in 13 years?), so not being at her sister's wedding is not a snub by exception to the norm.
Add to that the sister's apparent lack of graciousness in sorta kinda extending the invitation, and I think it's worth considering whether the OP would go to the wedding even were it on another date.
When I first read the original post, I thought perhaps the sister had chosen the date to allow OP to be at sister's wedding, and have the family together to also celebrate OP's milestone birthday on a date near the wedding - but, that certainly doesn't sound like the situation.