HiCustomerand welcome to JustAnswer,
It's uncommon for an oil pressure sending unit to be faulty, but it shouldn't be ruled out though. Even though it "acts" like an electrical problem, first thing you need to do is get an oil pressure reading. The dealer can do this for you. From experience, a flickering light usually means an oil pressure issue. It does sound like the engine is gelled or gelling. Once, it starts, the oil galleries gets plugged up and oil starvation occur. Another reason why I think the engine oil is sludged is because of the P0012 code. This code usually trips when the adjustable cam gear starves of oil. This gear needs oil to adjust timing depending on driving conditons. So, it sounds like inadequate oil pressure is present.
What you need to do is have the dealer pull the oil pan, valve cover and measure oil pressure.
If further inspection reveals gelling, should the engine get the full de sludging treatment as the dealer suggests to include replacing the valve guide seats or could cleaning the sludge from the pan, oil pick up tube and other visible areas be OK. It's 25 hours labor to do as they suggest. At any rate, the oil sending unit is fairly cheap and easy to replace, so I'll do that first. Could the tensioner cause the P0012. I'm really trying to NOT spend 2-3 thousand dollars. Thanks.
If gelling is found, a complete engine teardown will have to be done to see if the oil galleries and bearings are ok. Cleaning the pan and pickup tube will not take care of the problem. Once gelled, the gel is all over inside the engine. There's really no short cuts to fix it. A tensioner will not cause a P0012 code. I have not seen a failed tensioner to this date. The code is more likely from oil starvation as stated before. I understand you're trying to go the cheapest route, but to get it done right the first time, you will be saving a lot of headaches and money in the long run.