You are confusing a projected clause and an embedded clause. Clauses with embedded facts can be prefaced by "the fact/idea that"; they are not Complements of verbal processes but of processes of affect:reaction and the like.
He regretted (the fact) that he had lied. [affective process and embedded clause as Complement]
and then on the sys-func and sysfling lists at 21:51:
It's actually slightly more complicated than in my last as "the fact that he had lied" is not an embedding either. In this case the clause "that he lied" is projected by "the fact/idea" which is the (ellipted) Complement of the process of affect etc. So in such processes the Complement is an ngp.
In verbal processes with projections (cf. Verbiage), the projected clause is projected directly by the verb as process. In such processes there is not an ngp Complement.
 'Complement' is an interpersonal function, not an experiential one. The experiential function that corresponds to Complement is Verbiage/Range in verbal clauses and Phenomenon/Range in mental clauses.
 In SFL, the term 'affect' is used for interpersonal meaning, as a type of Attitude, from Appraisal Theory. The experiential function in Bartlett's clause is Process: mental: emotive.
 The fact that he lied is embedded — rankshifted — since it is a clause that is functioning as an element of the structure of another clause He regretted (the fact) that he had lied. See analysis here.
 Here Bartlett is confused by the distinction between fact nouns and projection nouns, discussed in Halliday & Matthiessen (2004: 468ff).