Thanks for the fast reply!
Yes, in that case, I do have polkit installed, and it's working just fine, because what you described is exactly what happens when Timeshift is started.
But even so, now that Timeshift is running as root, if one clicks Browse -- presumably, it's 'trying' to open pcmanfm, to also run as root, but the file manager fails to start.
I don't mean it fails to start as root, I mean -- it completely fails to start in any mode.
I should say, other Bodhi users are being (and have been, for some time) frustrated by the same problem -- everyone wants to be able to run Timeshift (properly, of course!) and there is no alternative, that I know of, that's compatible with pcmanfm.