You have various questions here.
If I understand correctly, roughly ~2.8 billion years ago cyanobacteria started pumping large amounts of oxygen into the atmosphere.
The great oxygenation event was not an instantaneous process and represents a change from one equilibrium (low oxygen) to another (20% oxygen). This abundance of oxygen and organic carbon means that there is "niche" for organisms that burn up pre-existing organic carbon and use oxygen as terminal electron accept. Fermenting organisms use the organic carbon as a feedstock, but don't have oxygen so the use highly reduced organic molecules as terminal electron acceptors (less energetic), making mixed acids (sweat bacteria) or ethanol (yeast).
Using modern industrial processes could this be emulated by making cyanobacteria farms to help slow down the climate problem?
Cyanobacteria and other aerobic phototrophs convert carbon dioxide to oxygen and organic carbon they can use to make biomass. So if you had enough of these you'd need to get rid of the biomass. Responsible forestry for housing actually ticks this box (cf. vs. a mature forest, which is nearly at equilibrium where the biomass gets consumed by other organisms, else you'd had a faster growing topsoil).
I also read somewhere that someone was able to genetically (or through other means) turn off the ability of a specific bacteria to reproduce causing it to produce a byproduct much more efficiently. Could this also help these "farms"?
It is a grand challenge of synthetic biology to stop growth. It is really hard to do as "cheat" (i.e. variant that can grow faster) will dominate. Biofuel research suffers from this as the fatty acid biosynthesis pathway is highly expensive energetically and cheats eventually win despite the most complicated safeguards.
Alternatively could we genetically strip the bacteria of all other functions other than procreating and photosynthesis? Making this bacteria live in a "bubble" would mean it wouldn't need a lot of defense mechanisms. This could make a single purpose bacteria that would be super efficient at doing this single task.
The carbon will be a problem, so you'd need to make biomass or some value added compound.
Three additional points:
There are many "green biotech" companies out there and there is an increasing number of green chemistry awards going to biocatalysis (enzymes for a reaction) and in some cases synthetic biology processes (whole enginereered organisms for a product or intermediate). But the former field is slow at growing because it needs to compete against a century of homo/heterocatalyis (normal chemistry), while the latter needs a lot more technological innovation to improve the production chassis (e.g. cheats, slow reactions, iron-sulfur cluster over-expression and electron balance are common issues: hydrogenase is a classically cited example of a hard target with great potential).
Note that most green biotech uncouple the process and use the biomass from regular algae or plants to feed their engineered organisms, as it is easier and there are less containment issues.
In the comments I mention, iron fertilisation of the oceans (a ship releasing a container size of nutrients across an oceanic crossing is make a substatial planktonic boom, which either increases fish levels or sinks and thus buries the carbon it converted into biomass). This is a curious case of huge scale and cheap algal farming. But we don't know all the details, yes we'd be turbocharging the oceans (which are barren relative to say a jungle), but we may favour toxin-releasing algae, jellyfish and imbalance fisheries and actually send to extinction others.
Bioreactor in the ISS
The Internation space station splits water to make oxygen, dumps all carbon waste outside (CO2 and ...) and has water and food shipped up. This is odd as you'd expect they would be the first to use algae to scrub CO2! However, things are changing: they are trialing an algal bioreactor, which on a larger scale will result in a near-closed loop system. However, there are many engineering and some biological issues that need to be faced.