You need three things top make the engine run: Air, Fuel, and Spark.
Any one of these could be the issue, but the symptom points first to fuel.
First - drain the fuel completely and replace it with fresh fuel and add about 2oz of seafoam to the tank. Run it a few times and see if that helps. (If this helps at all it may be just a temporary fix.)
Next, you need to investigate the following:
(in no particular order)
Dirt or debris in the carburetor passages and/or bowl
Partially obstructed fuel lines
Partially obstructed fuel pickup in the tank
Dirty or plugged add-on fuel filter (if there is one)
Gas cap not venting properly
Weak fuel pump
The Kohler manuals in the technical section may be of assistance.
Doing a google search for something like "kohler engine dies after a few minutes" is another way to find resources to aid you
No one will be able to diagnose it for you over the internet -- you will have to go get your hands dirty and do the investigating.
But by all means continue to ask questions if you get stuck.
If it is choking out as you say, The float and float needle are suspect. Brass float rarely spring a leak but it does happen, then they lose buoyancy. Fiber floats can absorb fuel with the same effect, they can also swell and widge in side of the carb.
When lack of fuel is the suspect problem, the first thing I do is at the fuel pump remove the fuel line coming from the tank and lower it below tank level into a container to check fuel flow for a minute or so. Poor fuel flow shows you need to work back to the tank. If good fuel flow reconnect the hose, then remove the hose at the carb. Put the end in a container and crank the engine over to check the fuel pump for good flow. Bad flow = bad pump, good flow = time to tear into the carb.