"overweening swagger" vs. "overweening pretension"

If you look at examples of usage of overweening at Lexico, you will see that it is primarily used with such words and phrases as

pride       confidence       ambition       sense of self-importance       arrogance       hubris

What they have in common is that they describe a psychological trait, an inner disposition.

Swagger, on the other hand, usually denotes external manifestations of such dispositions:

         an arrogantly self-confident way of walking : an act or instance of swaggering

         arrogant or conceitedly self-assured behavior
         ostentatious display or bravado
              (Merriam-Webster)

In contrast, pretension means vanity (Merriam-Webster), and so it also refers to an inner attribute. This is why it is a better fit.

Was going to suggest ambition as the best fit, until I saw the list of allowed alternatives.

Of those pretension is the only one that feels satisfactory in context. You need an attribute of character, which rules out 'wealth' and 'swagger' is more suggestive of behaviour than personal nature.