I probably should replied in the comments, but there is not enough character space. You asked if there is a cheap/easy way for you to step down a high current DC to DC converter. The word "high current" means expensive, and without good electronic design and fabrication skills, it will be difficult, if not impossible, for you. The easiest method is a linear 3-pin regulator with three 2N3055 transistor pass elements and a Zener diode. The most complicated method is a Buck converter. Good luck building either one.
So, now I will take off my engineering hat and put on my southern KISS engineering hat. One of my hobbies is flying electric RC aircraft; I am a 3D plane pilot, turning beautiful, expensive balsa wood planes into toothpicks. I would take one of my 3S (11.1 volts), 2200 mah 50C Lipo batteries, cut the connector off, and replace it with one that will mate with the lawn mower battery connector. If it were me, I would buy a new electric lawn mower. Another option is to look at the idiot's bibble (You Tube) for a solution.
Edit: Lefty, I watched the video and have a better idea of what the guy was doing. I looked at what you called a buck converter and instantly knew what they were. You see many on golf carts and trucks with 24-volt battery systems to power the 12-volt toys like a radio/stereo. You also see a lot of them in the Marine and RV markets. Your drill motor battery Is not a stiff enough source to worry about burning up the converter or lawn mower starter. You can find the converters at truck supplies, golf cart suppliers, and, of course, on Amazon for $20 to $30,
Your other option is to take the shrowd off the mower like the man did in the video and use your drill to start it. Good luck.