Although I can only offer as reference little more than 1) my age (having been born before the first appearance in 1959 of the word “jerry-rigged”); 2) a link to a likewise unsupported, Andru's zero-vote answer to the ELU question that you cite; and 3) my love of “real” jazz (i.e., the improvised kind that can never be played the same way twice), I am of the opinion that “jerry-rig,” with its pejorative “bâclé” connotation (as mentioned by Paul Picard) is nothing more than an erroneous, “garbled” version of “jury-rig,” with its wonderfully positive “improvisation” connotation (also mentioned in Paul’s good answer).
Therefore, based on my pedantic-like opinion of the term “jerry-rigged,” I’m going to conveniently transcribe the “jerry-rigged” portion of your example to read as “jury-rigged” and attempt to answer your question accordingly.
Perhaps again showing my age, our children grew up watching the tv show “MacGyver” with us and the title character of that show was undoubtedly the king of “jury-riggers,” and, although always ingeniuosly improvised, MacGyver’s work, IMO, was NEVER shoddy. In English, the word “MacGyver” has even been coined as a transitive verb (to assemble or repair something by ingenious improvisation, using everyday items that would not usually be used for the purpose) and “jury-rig” is listed here as its sole synonym.
My French wife, when asked how she would describe MacGyver, the character, and his improvisations in French, immediately came up with the “le roi du ‘Système D’,” with “D” meaning, depending on the audience, either "débrouiller" or démerder, from which one could get the adjective "débrouillard(e) (cf. démerdique?) to use in your example wherever “…..-rigged” is used.
On the other hand, the “jerry” part of “jerry-built” does legitimately imply shoddy workmanship (and, again without references to provide, I personally think it comes from the connection of the word “jerry” to “chamber pot” = “crapper” = crappy/shoddy workmanship), which makes the pejorative “bâclé” suggested by Paul a totally suitable translation of “jerry-built” in your example (although its use in the same sentence with “even” and “….-rigged” indicates to me that its author was born after 1959 and is a fan of neither improvisation, MacGyver, nor “real” jazz!) (cf. the expressioon: "close enough for jazz" [last paragraph]).
