Ugghhh...where to start.....?? Going to be some rambling in my post so you are warned....
P0605 is a communication error between ECU and BATTERY. Kubota words their stuff differently than auto manufacturers so in my opinion, forget what the ISO description is and use Kubota's description. It is more accurate. ISO descriptions are meant to be used in the automotive world and a lot of them don't apply 100% to Kubota's terminology--thus ends up with mass confusion. P0605 is "usually" a voltage issue. I forget what pin, but several of the pins in the harness supply +12v and "earth" (ground) to the ECU. If one of the pins in the harness end, ECU end are damaged/corroded/broken, it'll throw that code. Additionally if there is ANY damage along the harness somewhere that would kill the +12v supply (and/or earth), you will also get that code-and possibly others.
The only way to diagnose a Kubota system is with their diagnostic software. Nobody else's communicates properly. Kubota calls it "diag-master". It works well but like all software, isn't perfect.
Yes reverse polarity can kill the ECU. Yes the ECU is expensive. Yes the ECU is plug-and-play, so long as it is the exact part number for YOUR tractor. Some tractors have multiple different part numbers depending on the serial number and model. For instance the ECU is different between an L3901HST and an L3901DT, just used as an example. The "box" that the ECU is contained within is the same metal box, however the programming is different. The program has a part number if one were to reprogram the ECU (which was done way back in 2014/15 as I recall).
If for example one were to use a DT ECU with an HST tractor, some stuff won't work properly and then you'd have a massive waste of money/time. Similar for wiring harnesses, they are different between HST and DT (and GST or whatever).
Be really careful when ordering.....that is my only advice and that comes from experience. The part number that you are buying MUST 100% accurately match the one you removed. Dealer parts guys & gals are imperfect too, and that's where my experience came into play.
Kubota "might" assist with the repair....but don't get your hopes up. But it don't hurt to ask.
One final rambling. On all of the common rail engine tractors (basically everything +25hp), if you have to replace the battery, turn the key off an wait about 30 minutes before removing the battery cables. The reason being, one a lot of the smaller tractors, the ECU's capacitors will hold juice in them for a while after the power's cut. The ECU is still working when you turn the key off. Also you'll hear the electric lift pump still pumping for a minute or two after you turn the key off. It's pumping fuel through the supply pump and some of the lines and cooler, for lubrication and cooling purposes. Wait a while. On the bigger M series tractors, you have DEF in the equation. When you turn the key off on a hot engine, DEF continues to circulate through the injector which cools it off. If one were to disconnect the battery immediately after shutting the engine down, it's possible that the DEF injector could be damaged due to excessive heating because it's coolant was cut off. Just let it sit about 30 minutes before removing the battery cable; problem averted. Same principles with trucks & cars and they tell you in the owners manuals what and what not to do, and sometimes they even tell you why. I am not sure Kubota says anything about it.