I just checked my Madcow Super DX3 openrocket simulation. I added a 1kg mass near the nose cone. About 10 different motors got me to 3300' and above. All J motors which would be lower than the 2400N-s upper limit mentioned. I guess you could use a smaller diameter, lighter rocket to get 1kg to 1km, but I suspect that the 4" diameter is pretty close to optimal.
If you did use the Super DX3 or a scratch-build similar to it*, you could get your L1 and L2 with it (or do your L2 with a larger, heavier rocket to keep it low and slow) and then launch the 1kg object to 1km.
One thing. You mention "deploying" the 1kg payload at 1km. If you do that you MUST have a deployment system (chute, wings, whatever) to keep that object from coming down at velocity. Deploying a lawn dart at that height, to see how deep it penetrates the earth, is not an option.
*"To not know what has gone before is to forever be a child". That is, there is no sin in borrowing from the learning of others.