Pulling the board could be expensive. Some boards can be pulled hot, most are not. Repairing the board may not be an option, unless you have a lot of electronic experence. A lot of circuit boards are propitiatory now days. Even going back 4-5 years.
Who was the rocket scientist that connected the filter to the power supply to the a/c? Do not let him touch your system again.
The power, VA, available for contol power transformers is pretty limited. With out knowing specs for all of the equipment it would be impossible to guess what exactly is the matter. Gut says your air cleaner needs its own voltage and current source. You could always try the smoke test, put in a bigger control transformer and see how long it takes for the magic to leave...
Humor is applied to response.