I've done some searching around the forums and couldn't find anyone describing my exact issue, so here goes:

'04 Mini S with about 160k on the odo.
When COLD the engine has a tick. Not a rattle, or chatter, or clatter. I've heard BAD timing chains before and this is NOT it. (at least as far as death rattle)
It is a very linear and consistent tick or tap that increases in frequency linearly with engine RPM. People have said if the noise goes away above 2,000rpm then returns below 2,000rpm it is likely the timing chain tensioner stuck or binding. I tend not to rev it that high while it is ticking.
The noise goes away after a few minutes, or when warmed slightly. The time it takes to go away seems to vary with ambient temps. It is cold as h*** in the PNW right now, and I don't remember the sound being so prevalent in summer. It can take up to 3ish minutes at idle for the tapping to stop.
The noise is NOT present during hot start-ups, or while running the engine at operating temperature.

I have all the tools/parts necessary to change the timing assembly if I need to, and I had the SC PTO gearset for the water pump replaced about 20k miles ago (I heard a sound similar to the many videos of timing chain rattle. I inspected with "stethoscope" and the noise sounded like it was coming from the front of the engine. Took the SC off and spun the pulley and heard clacking. I had a local tuning shop rebuild it for me. Death rattle noise went away. This tap is different).
After taking the valve cover off and doing some inspecting (without turning the engine over) I see no visible damage to the cam sprocket, or the tensioner rails, and the chain is tight. I would like some info on how to diagnose a bad chain/tensioner
Should I remove the tensioner piston and inspect the guides closer?
when probing with a plastic trim tool I feel no real play in the assembly.
However a few of the rockers have a bit of side to side play. I don't know how much is normal, or if I should check some sort of spacing
Again, the taping is not present at operating temp, and usually goes away fairly quickly, but it can take a few minutes.
Noise has also been present across multiple oil changes

Edit: checked torque for rocker arm bolts. one of the 10 was slightly loose, and I was able to rotate it slightly before torque wrench clicked. all others checked out at 21 ft-lbs

 

Sounds as though you may have an oil starvation issue since the noise occurs only at cold startup and disappears when warm. I'm going to assume that your oil levels read correctly. However, I have read that different dipsticks may incorrectly read the oil level. A simple solution may be to add a bit of oil (past your full reading) and see if that makes a difference.

 

It is the original oem dipstick. I'm changing out the laminated oil cooler anyway. I could try slightly over filling.

 

The symptoms suggests to me the noise is not from the valve hardware but from an exhaust leak.