I love this trick! instead of lay on side i just had to teach this trick as "Bang your dead" Trick. This may be hard to write in words but I'll try my best. When you have your down laying in a down position, check his back legs if the dogs legs are on the ground in a ready to pop back up way, you wouldn't be able to lure the dog onto his side.
This is what i mean (each leg on one side ready to jump back up into action)
First step with the dogs down is to have your dog in a relaxed down as in the dogs legs are on one side of his body. This is a picture of what i mean with the legs (Dexter has his legs in a relaxed down both legs on one side, then theirs my super senior Libby breaking her stay tehe !)
You can try and lure the dogs legs to one side by using a really tasty food lure. Take the lure and try to kinda draw a circle around the dogs head in slow motion. that generally gets them to shift into a nice relaxed down with legs on one side.
Once you have them in that position you can lure them onto their side. Some dogs do not like this position because its very submissive, and it can be uncomfortable. I would try this on soft comfortable surface at first, with little to no distractions around. You can start luring at the dogs shoulder on the side that the dogs legs are resting, slowly take the treat and guide it around their head, if you do this really slow and let the dog follow the treat with his nose, maybe let him lick the treat a little you can slowly help them shift their body onto their side. This may take a few times for your dog to understand , if your having a hard time with the lure you can always try and tempt with a belly rub or a soft push, just make sure the second you get your dog in that behavior you mark and reward! Then have lots of fun with it!!