Light from sources at cosmological distances is redshifted, and we have several excellent lines of evidence for that, notably https://en.wikipedia.org/wiki/Lyman-alpha_forest which clearly shows that the emitted light was redshifted progressively as it reached intervening clouds of hydrogen on its way to our spectrographs.
Energy is directly proportional to wavelength. For light, E = hf, where h is Planck's constant. It's inversely proportional to wavelength, E = (hc)/λ, where c is the speed of light and λ is the wavelength. Redshifting means longer wavelengths, so less energy at the point of detection compared to the point of emission.
Energy conservation is local. At large scales, photons lose energy. Where does it go?
After you've thought about that on your own for a bit, compare your thinking with working physical cosmologist Sean Carroll's blog entry https://www.preposterousuniverse.com/blog/2010/02/22/energy-...
The connection to what I wrote above is that General Relativity guarantees that at every single point in a general curved spacetime there is a small patch of (quasi-)static flat spacetime. That patch can be ultramicroscopic deep within a black hole, or very large (in human terms) in interplanetary space in our solar system. In that flat patch energy conservation holds, as Carroll described. Outside that patch, we can see light redshift or blueshift through spacetime; the same energy shifts happen to massive particles like cosmic ray electrons, protons and neutrinos, too. It also happens to gravitational waves.
Event horizons break time-reversal symmetry. Nothing comes back from the other side. Generically, in the presence of a relativistic quantum field (like the 17 in the standard model of particle physics), an event horizon radiates greybody radiation comparable to a blackbody with a temperature inversely proportional to the horizon area. To save words, we just use that temperature (and go back to a calculation of the exact spectrum when greater accuracy is required). Black hole event horizons are really cold. Cosmological event horizons are ridiculously cold. And really it's the apparent horizon that has a measurable temperature. There may not be an event horizon in finite time, but in many circumstances the apparent horizon is indistinguishable by experiment. We have evidence of matter crossing to the other side of these horizons (e.g. from tidal disruption events and black hole/neutron star collisions); we have decent upper limits on horizon temperatures for a small handful of black hole horizons. (And someday we will have precision laboratory evidence of Unruh event horizon temperatures).
If the universe undergoes recollapse, black holes will tend to merge into ever bigger black holes as galaxy clusters freely-fall closer together (and then interact with each other, merging into gargantuan elliptical things with silly large velocity dispersions generating all sorts of collision & merger opportunities). So you need some unknown extra step to undo the time-reversal invariance violation of black hole horizons. Have any ideas?
The bright side is that black holes hold so much entropy that you aren't in any danger of violating the second law of thermodynamics via recollapse, unless your idea does that accidentally. The entropy is easy to see in a Boltzmann sense. Given the "no hair" conjecture, a black hole is fully described by a small number of parameters. Those parameters are a macrostate. A microstate is a configuration of things that fell into the black hole imparting its mass and spin and charge(s)). Entropy is proportional to the log of all the possible microstates for that macrostate. As more matter goes into a black hole, its entropy skyrockets. (The increase in BH entropy absolutely dwarfs the decrease in vacuum entropy; since hard vacuum is a macrostate, and every reasonable volume of hard vacuum is totally substitutable for every other similar volume of hard vacuum, there's a lot of entropy there too. As we fill the vacuum with ejected stars, radiation, high-metallicity dust, and so on as we squash galaxy clusters together during recollapse, that's a loss of entropy. But still nowhere near as low an entropy as the original cold gas clouds that were around after the cosmic microwave background formed. So your idea that recollapse means low entropy is on shaky ground, especially if you are taking your "binary counter ticks back to zero" idea seriously. You can take some comfort that lots of very smart people who can do the maths and have decent physical intuitions (Penrose comes to mind) have struggled with entropy in a cyclic cosmology.
Finally, the Hubble tension at its very strongest changes the age of the the oldest galaxies from almost 14 billion years to almost double that, not whether it the universe is expanding faster now than it was between one and four billion years ago. It's not something on which you want to rest a theory of cosmic recollapse.