Ebola, here are some things to think about.
Lawnmower engines and simple tractor engines do not have oil pumps. They rely entirely on oil-splash lubrication from a "flinger" extending down from the connecting rod. I do not know how they prevent the main and con-rod bearings from blowing up. But, those are very low-performance engines, and they are not designed for long life. (I believe lawnmower engines are designed for a life of around 100 hours running time. In a car with average driving, that's only around 6000 km.)
If the bottom-end of the engine uses entirely rolling-element bearings, you can get away with a lot of things that you couldn't do with a plain-bearing bottom end. This type of design is common in 2-stroke engines, but there are also a lot of scooter and small motorcycle engines that have rolling-element bottom ends. One thing to think about is that the crankshaft has to be a pressed-together design, because there's no other way to put the connecting rod (with its rolling-element bearing ...) onto the crankpin.
If your bearings are designed for pressure lubrication, you COULD store up the oil in a pressure vessel and discharge that into the oil lines, but why? If you got stuck spending longer than usual staging before the race, you're in trouble. The volume and weight of the pressure vessel may be significant, and the oil you pump through the bottom end has to go somewhere.
Personally, for a two-banger, I'd either go with an oil pump, or use a rolling-element bottom end and splash lubrication. Don't forget about the cam-follower lubrication, and how you're going to lubricate the piston pin.