Any transformer you use needs to be listed specifically for use in 120VAC wiring subject to the National Electrical Code. Do not use some random transformer out of an electronics supply catalog, even if it is UL listed as a component.
One of the Code requirements is that high voltage and 24VAC must be separated by a divider etc.
Generally these transformers are readily available in forms which solve that problem. For instance, a common form is a transformer built into an electrical box lid. The low viltage terminals are on the outside of the lid, solving the separtion problem.
I see these often as 2-gang box covers (4”) and I swear I've seen them in a 1-gang (2”x4”). We're not a product recommendation site but your question is do they exist; the answer is yes.
Phase issues
The problem you have adding a second transformer is phase. The two transformers will both be attached to R. If your transformer is out of phase with the furnace transformer, that means there would be 24v between your C and R, and another 24v between R and W when the furnace is off. So the thermostat would see 48V between C and W. You'd need to check with the thermostat maker to see if it can handle that.
May not need C
Some thermostats (perhaps not yours) can also scavenge the power they need simply by being in series with the two wires R and W. They do not need a C wire. However this trick only works if the furnace's control relay draws a fair amount of current. If it is too efficient, the thermostat can't scavenge enough power to function, and then they give you the advice to get a C wire.
So, you can fix this by making your furnace relay less efficient. I would say put a resistor in parallel with W and C on the furnace, except for that "must be listed" rule. So, use a furnace or air conditioning relay, which are readily available. Make sure your furnace's transformer has enough power to drive it. The purpose of the relay coil is to waste energy; the contacts are connected to nothing.
Fixed! Turned out, it was an issue with the condensate drain line.
I had moved it and it had become just ever so slightly raised, so that the line had become double trapped. The collector was filling up with water, and tripping the secondary pressure switch.
The reason I wasn't getting a code, was because the water was gurgling, or bubbling, as the negative pressure pulled the water back up the condensate line. This caused the pressure switch to fluctuate rapidly, which is also what caused the sputtering flame problem. This rapid fluctuation, which was picked up by a multimeter, was too fast for the control board to actually register a fault.
Fixed the condensate line, and all is well.
Haven't been able to explain the cover on/cover off phenomenon, may have simply been coincidence. It ran all night, all buttoned up, without issue.
Best Answer
I believe you misunderstood the instructions:
Sealing up the joint with tape/mastic after installing the bracket is part of the instructions. The installer omitted the bracket for whatever reason and just sealed it up. Could have been for noise reduction purposes as the other answer mentioned. Tape/mastic is not structural, it is all about minimizing CFM loss and making sure your conditioned air reaches its intended destination.
I don't know if there is anything particularly special about the missing bracket but you should be able to buy some sheet metal (28 or 26 gauge) from a local home improvement store or sheet metal shop and just fabricate your own bracket.
You need some sheet metal, tin snips, zip screws (1/2 inch self-tapping screws), 1/4 inch drive bit, and a drill. Just try to copy what is presented in the manual and make note of my image below. Avoid hitting the stuff noted in red.
If you want then add a bracket to the bottom and back side as well. Do not add a bracket to the front because those doors need to remain accessible.