@robberhands
I assumed the name change from Sue to Su is a joke. What else could it be? So I'd handle it as a joke and cede all spelling and pronunciation remarks.
You're right, in that it's not a legal name change, but it's requesting a new personal name. As I stated, it's because the characters are attempting to change their personas (i.e. how they're perceived by others). But in this case, they're trying to change the image they project, so they can change their reputations. As such, they're requesting their family's support in reminding them to act in accordance with the new role they're seeking. They won't insist on the name change with casual acquantances, or even most friends, as it's their image in the world they're changing by how they act rather than forcing people to approve of an odd-name choice.
It's a minor sub-plot about personal redemption in a story about one character helping others turn their lives around, so the story context (hopefully) makes all the difference, as readers see the progression that led to this change, underscoring why it works.
Given enough time (a couple weeks/months), they can safely drop the 'cutesy' names as they've had adopted the personas and those around them will recognize their personal change (i.e. more dependable, less flighty and irresponsible).
P.S. More to the point of your suggestion, they're not asking everyone to call them by that name, just certain key individuals as a 'reminder' to act consistently as their personal behavior changes are fully accepted (by them). Thus they won't repeat this mantra often, while at the same time, it's a personal favor, so they also won't belabor the point go rambling on about where the 'name' originates, how the pronunciation differs, and where it derives from culturally.
'Nuff said about a minor technical point. It's officially been beaten to death. ;-)