So, when I ask about DE arising in BBT, does it predate the origin of the CMBR?
Oddly enough, it's a yes and a no answer.
Einstein became convinced, to some degree, that there had to be some energy-force that was inverse to his GR view model of gravity. He had held the mainstream view of the cosmos, namely that it was static - no changes, perhaps infinite. But cosmology, I think, wasn't even a field of study since it was more than a decade later that the universe was found to have more than one galaxy in it, thanks to Hubble and the new giant (100") Hooker telescope. Einstein realized that the masses would cause a static universe to eventually collapse, hence the repulsive term (ie Cosmological Constant - Lambda) was included in his GR equation in 1917 (about a year or so after his 1915-1916 GR papers). [As mentioned earlier, this was a weaker net force since it was there to hold the universe, not accelerate the expansion of it.]
As you know, the beginning of BBT comes from Lemaitre, though Friedmann broke some ground with his math solution of GR several years earlier. It turns out, apparently, that the Lemaitre-Friedmann model includes Einstein's lambda term, but at 1/3 the value, surprisingly.
[see
here]. But its strength, I assume, is too little in the early periods relative to the mass density (ie gravitational strength) to be much of a factor. As density decreases, hence less local gravitational strength, this inverse force (now labeled DE) becomes more important. [To be clear, I think the modern lambda term is more of a constant (perhaps because it's called a constant.

) so it becomes important only when its strength becomes significant
relative to other forces.]
The math/physics is too deep for me to know whether or not something like lambda pops out of some solutions to GR, though I have read one account claiming it does, somehow.
As for DM by the early models of BBT, I can't imagine they would have included something that didn't become prominent until Vera Rubin's work culminating in her (et. al.) paper in 1978 [
here]. Even Zwicky's early strong hint wasn't until several years after Lemaitre's paper. So, DM would have become important to any BBT version after this time, especially when more studies were done on clusters, which would greatly advance Zwicky's 1933 study on clusters.
Apparently according to your answer, science does not know or if DE operated during the inflaton field epoch.
But, again, the Inflation model was anti-gravity gone beyond wild, so the influence of DE was about like asking if an extra flea slows a running dog. This goes back to the above statement that DE was no stronger in force per unit volume than it the value it has today (ie constant).
When it comes to DM and how much is needed to form structure and not say collapse the universe, looks like another fine-tuning project emerging.
Good question. Add to that just what is the gravitational equation for it. If you find one, I'd like to see it. All indications, and observations (e.g. Bullet Cluster), seem to show it isn't the simple inverse square law we know for gravity, but it's behavior is more sluggish. My guess is that a decade from now, a great deal more will be known of DM. DE won't be so easy, I think.
"The H-L tension will, very likely, get resolved, possibly with the help of the JWST." My observation, this is a hopeful response

Currently the H-L tension results in some very different universe ages appearing when examining using the cosmology calculators. So, I will wait patiently to see what develops with what I consider the many tweaks now needed to prop up BBT.
You're following this better than I am. These two values come from different time periods. The CMBR reveals the lower rate, and the study of galaxies today gives the higher value. But the Ho value is defined as the expansion rate today. So why does the lower value found from the CMBR study not make sense to actually be lower since, again, the greater mass density would find a slower expansion rate? Not to mention that with acceleration one must necessarily have a slower rate in the past. Does any article explain this issue? [Calling SPACE.COM!

]
Why did I do this some may think or ask? I feel tweaks like this in BBT should be fully disclosed to the public concerning cosmology and the answer to how the universe appeared and evolved into what we see today in astronomy. If many or a variety of fine-tuning problems appear, show it all and show the tweaks, including physics like the inflaton field in nature not seen today.
A very detailed account is given in Peeble's book, "
Cosmology’s Century: An Inside History of Our Modern Understanding of the Universe" (June 2020).
IMO, all this is normal astronomy. Recall that the first efforts at distances, which soon led to the expansion rate, found that stars were younger than the universe! This was because Hubble was unaware that Cepheid's came in more than one flavor. [Odd that this isn't mentioned very often.]