Yes, you can grab a sketch that will turn an arduino into a logic probe or analyzer, wire it up, and use it. I don't know how accurate or sensitive it would be. I do know the little 8 channel USB ones work, and can be found for less than 20 bucks. They can also be used as digital storage oscilloscopes, so if you don't have an oscilloscope they are definitely worth the purchase price right there. If you do much tinkering on electronic devices, you absolutely need an oscilloscope.
As for your "option 1" solution, the only way you could reprogram the existing chip (if it is programmable- some are burnt like a DVD or bootstrapped from a prom) is sort of like the ftdi method with the older arduino boards. You would have to have a complete programming studio for the chip in question. Not easy, plus you would need to come up with a sketch for the micro to work.
If you wanted to completely remove the micro from the main board, you could. Doing measurements while the micro was still in to figure out what signals come and go on what pins would save a ton of time in decoding the board. Using this method, once the stock micro is removed, you would have lots of easy to solder empty holes that would most likely directly plug and play with jumper wires and a generic dev board.
Going at the end of the trail is even easier. Letting the stock micro do its business and "jinxing" input and output signals is easier than writing a sketch to account for every possible situation. You could tap into certain points on the main board to keep tabs on signals and states of important aspects and ignore pretty much everything else. In this rig, the dev board would just sit and listen, allowing the factory micro to control everything while the unit is working like you think it should. When a condition came up where you wanted to change functional state, the dev board would either feed the factory micro a command to force the desired state, or switch output signals directly and feed the micro a signal to hide the change. Depending on how much redundancy is built into the stock micro, it may or may not go postal when its control is severed. Since a hrv is probably not programmed as a critical system, I'm thinking it wouldn't complain much.
Looking at the stock micro datasheet reveals it has a built in serial interface. The thing wires up using TWI (three wire interface) standards. It is 5 volt TTL and its pinout looks like this:
If the board is using the built in serial interface, you will see activity on pins 11, 12, and 13. Tying a logic analyzer to your wires between the main board and control panel, along with these 3 pins would quickly reveal whether or not (and how) the micro is using that port for comms. If not, other pins could be tapped to find out pdq which port is being used.
The easiest way to snoop on your micro is to grab a "piggyback" connector for your particular micro footprint. Most clamp on over top of the ic like a clothespin and have wires or sockets that you plug your analyzer into. Again, you can spend as much as you want on one of these. The expensive ones are more securely attached and are less fragile than the cheapo items. Depending on how clumsy you are with tiny probes and chips, a more expensive one would greatly reduce Murphy's law from showing up and ruining the day. Even then, you still have to be careful with static electrical charges and shorting pins and such.