Well your'e in uncharted terrority here, you essentially have a "frankenstein" engine. Not that this is a bad thing, I know UD has zero support for the older engines, and many guys have no choice but to buy a hino engine and swap parts over. Usually they perform the swap with a engine that was built in the same date range. I have never dealt with anyone who has used a engine that was 9 years newer. I also have limited experience with UD.
This is what im assuming you did, because I dont know how else it could be done. In order for this to work you must have removed your UD engine, stripped it to basically a long block minus the injectors. Then purchased a used 16 hino engine and trans, you would have had to essentially installed all of the old external components from your ud on it. The oil cooler is not compatable, the air compressor coolant lines and the egr coolant lines must have been changed. I have no issue with this but it creates a can of worms so to speak.
At this point I would eliminate the simple items first before we condemn a head gasket, or air compressor failure.
I would like to be 100% sure on the radiator cap pressure ill double chevk the hino spec for a 16my engine.
Its very curious that it only pushes after the engine is off. To eliminate the tank, and valve in the top of the overflow tank try this. Remove the line from the nipple that goes to the overflow tank, install a line into a clear water bottle, run the engine up to the temp it usually pushes coolant, shut the truck off and see if the water bottle fills up. If so how much coolant and does it appear to be contaminates with oil?
Where is there a small line from the egr coolet attached to the top of the radiator somewhere? This is the vent line for the EGR cooler and nees to be connected. Any pictures of the overflow tank, radiator, egr cooler and air compressor would be helpful in getting to the root cause of your issue. Thanks