Today, I found out that the very common beat pattern that I mentioned in my answer has a name. (I knew it had to!) The section that is confusing you sounds like the Tresillo rhythm.
If you search this website for that phrase, you'll find a handful of posts that mention it. I recommend you look at these Q&A: What is this beat and why is it so popular; Different way of “grouping” eighth notes. They are describing almost down the T what you are playing in that section.
Original answer continues below:
How you intend to notate this depends on how you intend the groove to feel. I listened to your audio sample, and the best indication here is that your problem is merely with the metronome.
The rhythm of the first eight bars is on the left; the rhythm of the remaining bars is on the right. The difficulty you had in describing it notwithstanding, the pattern you are utilizing is extremely common in contemporary music. It's just about everywhere.
What sounds, to you, like 5 triplets with a partial triplet at the end (or however you wish to describe it) is almost certainly that figure on the right. The accents account for the "5 triplets" sound, and the final beat (of 4 sixteenth notes) of the bar accounts for the remaining "1/3" (that still is confusing to me). The reason it "lines up" after several measures is just basic math. I probably don't even have the math correct on the tempo, just be forewarned. I guessed. I do music, not math.
This gets back to the groove.
I found myself nodding along. I enjoyed the sample. I grooved. The metronome was just keeping time in the first 8 bars. Thereafter, it just killed the groove. Attempting to fit your groove to a strict timing structure of the metronome (in this case, calling it some Frankenstein's monster of 3/4) is asking for a headache---or, asking on Stack Exchange. For what it's worth, if you didn't have the metronome playing at the same time in that audio sample, and if the metronome weren't to be considered at all, the rhythm described above is what almost everybody here would conclude.
It is likely going to be much easier to figure out how to change time signatures in your DAW than to erect some convoluted solution to the timing of your music that requires it to be in 3/4 the entire time without changing the tempo.