Hey @James_Tuxbury!
I think you can do this via CSS for the text-style when the tooltipText property is not null. You should be able to use component values in the custom CSS (Inject CSS class via JS - #3 by Jack_T)
Alternatively, you can just style the text conditionally with Markdown. You probably can't get the dotted underline this way, but you could do some other things like include additional characters/icons or even just make it look different slightly with other Markdown formatting (example below):
No tool tip:
With Tootip:
Mostly just showing how to deliver the differing styles and messages which can maybe be useful to you.