Regarding the lack of interest in Bergson during 70s and 80s, I think you are precisely right, and the untestable nature of the time-like ramifications of relativity weren't something I had previously considered. Of course, by that time Einstein was so obviously right, and Bergson so obviously wrong, I think those physicists can be forgiven for not knowing, or for not giving a shit if they did know.
One of Bergson's chief objections to the Twin's Paradox was the idea of time slowing down for the twin sent on the relativistic journey. Such a thing made no sense to him, giving how he framed time: as an unrolling now that could not be subdivided into metric units.
Bergson's objections to time-like relativity are certainly understandable, I think, given the historical context. As you pointed out, the notion of a physics without a background of absolute space - the concept of the ether or an absolute background metric against which space time is measured - were the 'standard' model of the time. I would go even further, and say that many physicists at the time either had severe difficult in coming to terms with physics based on frames of reference, or they rejected it outright. So I don't think Bergson's objections to a relative experience of time are unreasonable, nor do I think you can fault him for his objections, given the difficulties physicists themselves had coming to terms with the implications of relativity. Something I hadn't really considered, however, is that we didn't have the laboratory apparati to test the hypothesis that time passes differently under acceleration until decades after Bergson himself was dead.
Regarding clocks, I certainly understand that a 'clock' in physics is a shorthand for a physical system undergoing periodicity: whether it is an actual clock, a cesium atom, or a gas, etc. For Bergson, however, it was the act of reducing the dimension of time to a countable metric itself that was problematic. For him, the idea that time can be subdivided like space was simply a trick of memory, not actual experience. If we focus only on the unfolding 'now' - something difficult enough to do Bergson wrote whole books on it - we only see one moment elide seamlessly and smoothly into the next.
Bergson had no problem with pointing out that metric time worked quite well in modeling physical systems; his objections were to using this approach to model human experience (particularly with regards to free will and the implications of determinism inherent in relativity). Bergson was a proto-postmodernist, and was trying to get at the idea that the 'map is not the territory.' Hence Bergson's focus on the Twins Paradox. Relativity allows for a space-like time that can be 'run in reverse,' but actual time isn't space-like, in the sense that it can be traversed in one direction only. So despite what Einstein's equations predicted, Bergson objected that the notion of the Twins experiencing time differently was non-sensical.
What I hadn't realized prior to reading your comment is the similarity of Bergson's objections to the objections/difficulties physicists themselves had in abandoning the idea of a fixed, background metric space. He is essentially arguing for a fixed background of indivisible non-metric time that everyone experiences universally and that unrolls at a fixed rate for all observers.
On a side note, I've always thought Bergson (and pretty much the entire history of the philosophy prior to Einstein) had it precisely backward. Thousands of works have focused on and prioritized time as a cornerstone philosophical concept. Bergson was not alone is his obsessive focus on it. And yet, time is the most ephemeral and intangible concept of them all. You can't see it, you can't hold it, there is nothing there. 'Time' as we know it is merely the periodic spatial change repetition of some physical phenomenon: the vibration of an atom; the periodic steps of a watch hand; the filling of a fixed volume of space with water (as in a water clock).
Perhaps it's only the fact that I take living in a post-Einsteinian space-time for granted, but I always found it strange that people -including Bergson - so obsessively abstract 'time' as something distinct from itself, when what they are really seeing is space itself unfolding into... well, more space I suppose.
Thanks again for the thoughts, it was a great read with my morning coffee!