Does it make sense to "reorder" a categorical feature to make it monotonic?
I would not recommend doing this in general as it will likely lead to overfitting.

While a particular reordering might improve fit on your dataset, the goal is to create a model with good predictive power. If you received never before seen data from an independent second, third, one hundredth Titanic, would the ordering still be the same? I doubt it. You'd probably be incorrectly using the somewhat arbitrary ordering fixed on the first, real Titanic on new data where the ordering is no longer the same.

I think the story goes that the young and the elderly got preferential treatment and this is why their survival rate was higher. Therefore your AgeBand curve should be approximately U-shaped. (Which it is, approximately.) If you insist on linear regression, I don't see how you could ever (directly) include this nonlinear (U-shaped) effect.

The fewer assumptions (particularities) go into the choice of features, the better. It would then make more sense to me to reorder such that you pick bands alternately from the two ends (starting from the middle): 3, 4, 2, 5, 1, 6, 0, 7 (or in reverse order, starting from the ends). Or, even better, pool the age bands into coarser groups: {1,2,3}, {4,5,6,7}, {0}. I think these are justified by your exploratory data analysis.

When I look at the range of values in the AgeBand plot ($0.33$ to $0.55$), this variable does not actually explain as much as Pclass, Sex, FareBand. In the end you might end up dropping it from your variables anyway, so once again, the more attention you pay to it, the more you're at risk of overfitting.