I think the important thing to keep in mind is that the electron releases the energy it absorbed to fall back down into its more stable energy state based on its original "velocity". So you're not exactly wrong to suggest it was traveling at a faster "velocity" than it should be because maintaining this velocity would be unstable. That said, physics at this scale makes any intuition difficult, if even completely accurate. We assume the best intuition based on science that works, while simultaneously knowing it is a lie. For example: the plum pudding model was entirely reasonable and useful, until it wasn't.

I think it is important to consider how properties of waves intervene when the billiard ball analogy breaks down. I know it is popular and practical to reference its probability distribution because quantum mechanics, except for the fact that anything can be reduced to statistics. It's helpful for solving a physical property we want to solve for, but it wasn't statistics that ascertained the fundamental property. It's a tool that leverages hindsight, and sometimes it's conflated with physical property itself and so understanding suffers. Imo.

Consider De Broglie and Noether. Noether convincingly demonstrated that for a given field, you have an associated particle. You can think of orbitals as "localized" fields of the same type that manifest differently based on the geometry of 3 dimensions, and by constructive/destructive interference. in my mind, this field is born from the vibration/"a kind of orbit with no center"/"a particle somehow maintaining an occupancy in space" of the nucleus.

The nuclear vibration isn't 100% symmetrical all the time (because environmental influences are unavoidable), and so it gives rise to perturbations (I.e. electrons) in these orbitals/fields in like manner to itself being a purturbation in an non-localized field. This explains why electrons "spin" instead of "vibrate", and also suggests that these particles don't travel up or down between orbitals, but trade orbitals without travelling to maintain symmetry and stable energy states all dictated by the nucleus.

This also suggests something very interesting: a stable nucleus with field orbitals and no electrons is possible in ideal conditions.

But to *actually* answer your question with regard to relativity: electrons have mass, and if you can associate mass with an energy state, you can solve for a "velocity" property for its kinetic energy. Quantum mechanics wouldn't be able to say much about the mass dilation (I don't think?), In the same intuitive sense. It would just say that for the probability of the electron's location, the electron increased in mass based on its energy state. I see it as a situation where QM and relativity aren't exactly compatible, but they don't exactly contradict each other.