1116.44fps at sea level. Generally it's rounded to 1100fps for simplicity. I know you are giving an estimation but it's a tad high is all.
I shoot sub sonic lead cast .32 pistol bullets out of my Lee Enfield using magnum shotgun powder. You may have to develop your own load to see what burns properly and also gives low enough velocity. Obviously it wont cycle a semi auto.
Due to the case volume difference you may have to develop different loads for both the .308 and .30-06.
Low volumes of rifle powder doesn't work as you don't reach a high enough pressure to burn it properly. I tried loading 20gr of H4895 into .303 British cases and it fired at about 800fps but loads of unburnt powder was left in the case. Load data for a smaller case at the desired velocity also wont work well since the increased case volume will mean less efficient burning.
My current load for .303 British is 5gr of Alliant Blue Dot but that is with a 90gr pistol bullet and it pushes it around 1000fps. The heavier bullet in question will produce higher pressures with the same powder loads as well as travel slower.
To develop a load I would start with a slow shotgun powder (HS-7, 2400, Blue Dot, or maybe even something like H110) at around 5 grains. First shot check for overpressure signs but chances are you will be under pressure. I get a lot of soot on my .303 cases because there isn't enough pressure to expand the brass to seal the chamber properly. Also make sure there is a new hole in the target or else check your bore to make sure the bullet actually left. Then go from there with different amounts of powder to get the velocity you want.
To clean soot out of my chamber I use a .410 bore shotgun mop and 3" cotton patches with a touch of nitro solvent. Make sure to dry your chamber really well after though.
I also load .45-70 with 500gr lead cast bullets over 16gr of Blue Dot and get around 1150fps. It just barely breaks the sound barrier and I'll be reducing it slightly to get below it in my next tests.
This site is a great help for me:
http://www.gmdr.com/lever/lowveldata.htm
It shows for a 170gr lead bullet in .30-06 and Blue Dot powder (the one I have on hand and use most often) to be 7-11 grains. So with a heavier bullet 5-6 grains would be where I start. There are enough powders listed that it shouldn't be hard to find something you either already have or can get easily. At the pressures we're talking here blowing your barrel/chamber isn't a real risk. Having a bullet not leave the barrel is a real risk though so always check for that until you find something reliable.