Aussies?

[Tellos Athenaios]

FPTP has its real strength in continuity, as it tends to create 2 or 3 parties that share power, slowly absorbing ideas and change based on factions within those parties rather than trying to represent 1400 opinions in one legislative assembly. FPTP is, however, almost remorseless in screening out smaller/emergent political efforts. This keeps the whackoid fringers out, but makes change efforts much more difficult. The USA exemplifies both of these characteristics.

PR has its strength in inclusivity, with all voices actively represented in the discussion within the legislative body. This is also its weakness, however, in that too many disparate voices may have great difficulty in establishing a governing coalition of some form as the legislative body may be too balkanized, with the inability of a party to develop a workable coalition granting too much de fact voice to some of the more extreme or minority "parties" in the legislative body. An example of this would be the Knesset.

Neither approach is without value, but neither is without flaws.

Sure, though I would argue that FPTP always exhibits its worst flaws in any setting in which it is implemented whereas the flaws of the PR system only tend to matter in a political worst case scenario (there's a reason we call it Balkanisation, after all). Interestingly that is exactly the same case in which FPTP is most blatantly "discarding" votes and artificially propping up the winner, since in this scenario every candidate or party receives very limited real public support (e.g. gaining the majority of seats by achieving 30% of the votes or less, simply because everybody else does even worse). In FPTP you are pretty much guaranteed to see its worst side even in "fair weather" (since you routinely reject approximately 50% of the vote by design[*]), with a PR system this is much less the case.

Perhaps a bicameral legislative body with one body elected using each approach?

Why? No, really: what advantage does artificially increasing the political power of a few most powerful factions give you? Presumably there is legislative veto power involved, so now you are adding artificial incentives for obstructionism to combat a politically disfunctional situation. How is this going to help matters?

* Yes I am making some assumptions here, mainly that "shortfalls" and "excess" votes are relatively evenly spread and tend to cancel each other out. (E.g. for every seat gained by ~25% of the vote there would be a seat gained by ~75%.)