Different endings are not unknown to Dickens. After all, Great Expectations has two endings. One possibility would be that Charles, Lucie and their daughter were killed anyway. That would seem to negate Carton's sacrifice, and it would be sad, but it would demonstrate a more pessimistic or realistic ending to the book.
Carton could also have said to heck with it and left. It was a big commitment, after all!
This is one of my favorite novels of all time, and I can't really imagine a richer or more satisfying ending than the one Dickens has provided; however, I'll try.
Any alternative ending has to deal with Charles in prison and Sydney's determination to rescue him. It is also a given that no one else (except perhaps Lucy, which isn't a viable option for a mother) would be willing to trade places with Charles. Only a few things seem reasonable, given those two facts.
One possibility is that Charles is executed as planned and Lucy and her family go back to England. The unfortunate victims here (other than Charles, of course) would be Dr. Manette--trapped in his lost self again, probably for good this time--and Sydney, who would ever be on the fringes of this, his only family, suffering from unrequited love.
Another possibility would appeal to the romantics. Charles does die as planned; however, the family returns to England and, after time, Lucy falls in love with Sydney and they live happily ever after with the bittersweet memory of Charles between them.
Other, more far-fetched possibilities (such as Charles breaking out of prison or Sidney kidnapping Charles as he's walking to the guillotine) could be considered. Totally outrageous possibilities, such as Dr. Manette leading a counter-revolution, are not even plausible.
Most great novels, those which capture the best and worst of human nature and stand the test of time, already have effective and satisfying endings. Sydney's actions are the epitome of sacrificial love, and it's hard to improve on something so pure.
How about making Sydney Carton's vision of the future come true? Carton has a vision in which he sees that Lucie and Darnay will never forget him as they escape to England. Carton's namesake, Sydney Darnay, becomes a famous judge, fulfilling the career that Carton had always wanted for himself but could never achieve because of his alcoholism. Sydney Darnay goes to Oxford, becomes a laywer and then a famous judge. Let's give him a role in something super important for humanity - perhaps he can be a judge who helps William Wilberforce abolish slavery in England. Since slavery was abolished in 1833, this could work time-wise. Little Sydney can grow up to be a famous abolitionist, William Wilberforce's right-hand man. That way, the sacrificial death of Sydney Carton would have a long-reaching and positive effect on humanity, since he is a type of Christ figure in sacrificing his life for the life of another. It fits in with the theme, don't you think?
A black hole sucks the whole world and the whole story starts again in an alternate universe with two different cities: Dallas and Tokyo.