For some reason I was picturing the cooler with the fittings pointing down. Starting the engine with the filter off could be very messy, and I would not feel good about running the engine with no oil flow.
Thanks. I agree that starting the engine with the filter off could get very messy, which is why I'm thinking about how I can modify an old oil filter to direct the oil into my bucket to prevent a mess. As for running the engine without oil flow, I would probably only need to run it for 1-2 seconds to flush the old oil out, which isn't much more than the time it takes to fill an empty filter. But then the question becomes whether 1-2 seconds without oil flow is more harmful than a bit of used oil getting left in the system.
The other option is to disable the fuel and crank the engine over to push clean oil through the cooler rather than actually starting it. The other benefit to doing it this way is I'm not as likely to make as big of a mess since the oil won't be spraying out as fast and I can stop as soon as I see clean oil, so I won't waste much new oil. I will probably try this method first and only start the engine if the oil isn't flowing out at a decent rate cranking the engine over.
I actually ran this very same setup on my brother's old Pontiac Vibe 1.8L 1ZZ-FE Toyota engine that we supercharged. We had a blower on it that made an extra 70 HP or thereabouts, and we also used it to tow a trailer, so I wanted to ensure that the oil was kept at the right temperature no matter how hard we were beating on it. I had almost the exact same braised plate heat exchanger plumbed to the filter adapter on the oil side and to the heater circuit on the coolant side. Worked very well for its purpose.
But now to answer your question: Don't bother starting the car with no filter on. What a disaster that will be, and what a complete waste it is as well. You will not be doing the engine any good by trying to drain a few extra ounces of oil from the cooler and lines. Honestly, just change the oil as normal and be happy.
My experience with this setup was great. We ran this thing for years and years before my brother decided to get rid of the car and upgrade, supercharged and all, with zero issues. We changed the oil every 10,000 km, did not ever try to drain the extra old oil out or anything like that, and never worried about it. In all honesty, there is old oil left in the engine that can never be drained in every car, and no one worries about that either. So my vote from experience is to enjoy your new setup and don't risk running the engine without a filter. Not worth it at all and nothing material to gain from it.