I do not know how exactly the attitude is presented to the crew, my experience is tied only to the unmanned spacecrafts, though I would imagine there is no unique preferred solution and it depends on the particular task at hand. I will try to provide some possibilities at least, but first a slight correction.
You mention the rotation matrices and quaternions having unique rotation representations. However, that is only true for rotation matrices, while the quaternions have double covering of the rotation space. Every orientation has two quaternion values attached to it.
Also, a bit about the gimbal locks and singularities. Gimbal lock is a loss of a degree of a freedom in rotations, which is an actual effect of a physical gimbal. But a spacecraft is not physically tied to a gimbal, it is freely rotating in a space and will never experience a loss of degree of freedom of its rotations. On the other hand, some actuators for attitude control might experience it. Gimbal lock is not a problem with representations at all, it does not prevent from knowing the attitude in any way. It is only an analogy to the problem of singularity when Euler angles are used, same conditions apply for both to happen, but instead of loosing a possibility to do something, as with gimbal lock, the singularity removes a possibility to know something.
This knowledge which is lost is not a knowledge of the attitude, it is perfectly clear in which orientation spacecraft is when singularity is reached. What is lost is the path to it. Non singular attitudes have one single path to them from some reference orientation, but a singular attitude has infinite paths. So if the goal is just attitude knowledge, any representation could be used, but a careful selection is required when the attitude change is of concern.
While algorithms have their ways to evade issues with singularities, I see two possible ways for crew members to go about this, building the intuition about the used attitude representation and selection of the appropriate reference frames.
I am probably biased for the approach of building the intuition, as I have already spent enough time dealing with quaternions, but I claim that if you stare sufficiently long at them, you start to understand them, know them, feel them, form a quintet (quinternion???) with them... sorry, got carried away. Anyhow, the intuition can be improved and then move on using the quaternions directly. If you want to pilot a spacecraft, you need to learn a lot anyway, spacecrafts are not built to be immediately obvious and intuitive for everyone. Training is required and the training includes attitude understanding. This holds for Euler angles too. Crew gets trained to use them and they understand even when the attitude is in singularity. Pilots do have context of how the singularity was reached and they should know how to get out of it towards their goal. A sudden jump in values from -90 to +90 should not be big problem if expected.
Reference frame selection
Then comes a proper selection of reference frames. You do not have to stick to one and only reference frame all the time. In case of a flipover maneuver from the question, it should be possible to take current attitude as a reference. Then perform actions necessary to get to the attitude which is 180 degrees away. The singularity will not be on this path with Euler angles. If the representations is Modified Rodriguez Parameters, then the singularity will be exactly at this targeted attitude. So in such case, theoretically, you can just guide the spacecraft towards this singularity point, but practically that would be a bit harder. A better approach, for any used representation, is to make your reference frame your targeted attitude, and then the guiding becomes going towards the attitude of zeros. In this situation the Modified Rodriguez Parameters are a really good match, as the guiding becomes always going away from the singularity. Still, even if sticking to the Euler angles, there should exist appropriate reference frame to avoid singularities.