It seems to me that the most natural split-alignment where an inverse system is part of it would be along these lines:
present tense, non-perfect(ive), ... : inversepast tense, perfect(ive), ... : ergative
The appearance of the ergative part following the same lines as usual in split-ergative languages. However, we could maybe do something else. How about a scenario as follows:
A nom-acc or erg-abs language that has a reflexive possessor along the lines of Swedish "sin", i.e. 'his/her/its own' where his/her/it refers to the subject (or the absolutive, if we want hardcore syntactice ergativity in the ergative subsystem). We form a bunch of adverbs along the lines of "after doing VERB" and so on as "after hisrefl VERBING y", whereas intransitives don't take subjects: "after VERBING", and with the other subject we get "after hisnonreflexive VERBING y". We go on to grammaticalize these types of adverbs into predicatives. We lose the reflexive pronouns elsewhere, after first reanalyzing them as part of past tense agreement morphology. We further reinterpret the two morphemes as referring to whichever noun is higher on the animacy hierarchy (or other relevant hierarchies) rather than to previous subject/previous (dominant?) non-subject.
Now we have a nom-acc or erg-abs language with a good start at getting an inverse system rolling for past(/...) verbs. One thing that would help rid these past tense verbs from a more clearly nominative-accusative pattern is if non-finite verbs take nominative rather than accusative objects. This does happen in some languages for whatever reason so no weirdness there - except we of course need some rather specific circumstances for that to vanish. For an ergative-absolutive language, the problem with accusative objects steering the thing back onto an erg-abs pattern is not applicable.
However, since ergatives and genitives often are morphologically identical, it could initially seem unlikely for something like this to appear, as the non-finite past tense verbs would be parsed as having ergative subjects anyway - but there's the clever thing, the reflexive/nonreflexive distinction for possession might not really apply for pronouns-as-subjects but very well for pronouns-as-possessors. Or alternatively, they may be to distinguish whether it's the previous ergative or absolutive noun that is referred to in a new sentence (or subclause), and this - I would figure - increases the likelihood slightly for something like this to come about.