If it has been sitting in the clean new 30 weight oil already for quite some time. I would just exercise the pump and leave it in there. I do not think normal car based oil has an additive that will be destructive in itself. The spindal oil is a bit of an unknown to me and may have some additive package as well. Or perhaps not. The viscosity is going to be a lot less though.

Runaway Fail
 

I found out the hard way a couple days ago that a pump that has been sitting for years can fail where instead of not delivering enough fuel it delivers way too much and results in a runaway. The rack is stuck and will not move to the shutoff position. Of course it never occurred to me to open it up and look before starting the engine. Fortunately I was able stuff a rag in the intake to slow it down and disconnect the fuel line from the lift pump before it got too crazy.

Lesson learned- If you are starting up a motor that has sat for some time, be armed to deal with a runaway. I've since started the motor again and killed it right away using the block of wood over the turbo intake method. Then I started it and position a thick layer of cardboard over the turbo intake with just enough of a gap to keep it at a high idle. The reason I did that was I hooked up the fuel feed and return lines to a jug of Marvel Mystery Oil, diesel purge style, to circulate through the IP. Now the IP has been full of MMO since last night and in the next day or two I'm going to see if that has loosened anything up. I would be tempted to store a pump in MMO for long term storage.

why mmo, why not use actual diesel purge? that's what its for after all, is cleaning out the fuel system components.

Partly because I felt that MMO would do better at loosening up corrosion and that it would have better lubricating properties. Mostly because at 11:30 at night I had a quart and a half of MMO at the shop and no Diesel Purge.

5 gallons of spindle oil goes for around $150. Not cheap by any means. I'm guessing you need 5 gallons to cover something the size of an IP.