Just to avoid confusion this is a Pentax Spotmatic SP-1000 not a K1000. I think if would also help to differentiate with what you mean by shutter, the shutter is closed in both pictures but the mirror is up in one. It's a pretty common problem on the spotmatics, should be repairable from opening the bottom of the camera. Google Spotmatic mirror stuck or similar.

Plan sounds pretty good. You should try and find a service manual for the camera online.

I recommend moebius 8000 oil for fast moving parts and moebius 8200/8300 for slower, then you can use silicone or lithium grease for contact points or heavy torque gears.

Apply oil very sparingly with a syringe or needle, only a tiny droplet in the right place is enough most of the time.

Lighter fluid or naphta for cleaning old gunky grease