Mike - you may have something in the oil passage in the engine where you are attaching the tubing for the gauge. When shut down and cool, the debris settles. When you start up, the debris is pushed to the side of the passage and stays out of the way of the established flow path to the fitting/copper tube running to the gage. When you shut down warm, the debris is free to move in the warmer oil and may partially block the passage.

Have you removed the fitting from the block and checked for an obstruction? It doesn’t take much to restrict the flow to the flow/pressure to the gauge.

I’d also change the oil (warm) and see if that doesn’t take care of the problem.

Good luck on this,
Glenn