First of all, kudos for your gray matter being smart enough to build with a hub and split sprocket.
Secondly, I'll tell myself you're just not finished with the build yet. Since that shopping cart could hinder your son from being thrown clear in a rollover (no seatbelt when there's no rollcage), I'll tell myself the rollcage and 4-point safety harness are your next step in the build.
Thirdly, T.Davidson sees the same thing I do, maximum strain from the motor pulling on a sprocket in the middle of a long stretch of unsupported axle, likely flexing it under load. You need another pair of pillow blocks mounted as far to the inside of your frame plates as you can get them. Fourthly, you got it right with the split sprocket, but blew it with the solid locking collars, should have been
double-split collars, and
NOT just "split collars". You need one on the inside of both existing pillow blocks, and one on each side of both the sprocket hub and the brake disc. Have to pull the bearings off to add more bearings anyway, good time to get rid of those solid collars with the setscrews that are scarring up divots into your axle. Sand those out. Double-splits install in 15 seconds.
Now about that chain skipping teeth when under load.............Usually due to too much slack in the chain, sometimes due to axle flex, motor twist, or frame twist. You can't say a chain has .25 - .50 slack when chain is short as yours, and also when it's as long as my 3-footer, it's different for every length of chain, and I'm betting yours is too slack. Your motor has a solid mounting, minimal squishy pads under the bolts, and a short chain, no problems there. Likely has some axle flex, but you can temporarily tighten the chain enough to stop the skipping of teeth until you get another pair of pillow blocks on it. Check out the links below.................
UCP205-16 Pillow Block Bearing 1" Bore 2 Bolt Solid Base (2PCS) (jeremywellindustry.com)
1" Bore Double Split Shaft Collar Black Oxide Set Screw Style (4 PCS) (jeremywellindustry.com)
Jeremywell Industry (links above) has great parts and service, but you can get that kind of stuff from Amazon or Ebay a little cheaper.
EDIT: Don't forget about the existence of half-links (offset links) for making up chains when a full master link is too much..................
#35 Roller Chain Offset Replacement Links. 35-OL, P/N#76351, FREE SHIPPING | eBay