Hi. You've made good progress, but your workflow is more complex than it needs to be.
You can easily successively rename Images in any Images view (Browser view, Version view, Split view). Just show on the Metadata tab of the Inspector any Metadata View that includes the field "Version Name". Select in the Images view the first of the series of Images you want to rename, and then click in the "Version Name" field on the Metadata Inspector and rename the Image. Pressing "{Command}+{Right Arrow}" records the change you just made and moves the current Image selection to the next Image. Note that the entire content of the "Version Name" field will be selected, which makes it easy to replace (rather than amend) the Image name.
I believe the reason there is no keyboard shortcut for the single action is because the functionality is built into the UI.
Incidentally, there is a mouse shortcut for "Move Selection Right". It is on the far right of the Control Bar. (Show/Hide the Control Bar when a Viewer pane is present by "Window ▹ Show[Hide] Control Bar", or the keyboard shortcut for that command, which is, by default, "d".)
The interface is rich and deep and, imho, exceptionally well-designed. The materials which explain the UI are good — I recommend memorizing the first eight chapters of the User Manual — but still fall far short of allowing users to find answers to questions such as yours. I hope my answer is useful.
Your next question might be "How can I move step-wise through the Images in a multi-image selection? The answer is to use "{comma}" and "{period}" to move the Primary Selection left/right among the Images selected. This, however, will not allow you to rename successive Images in a multi-Image selection. For that, simply create a new Album from the multi-Image selection, and use the method above with the new Album as the selected container. Of course, you can delete any Album at any time without losing any Images or metadata.
HTH.
—Kirby.
(Minor changes made.)