I use Havoline dino 5w20 with LC20 because my oil life monitor indicates service due at about the 6000 mile mark. I will get a UOA to fine tune the LC doseage I need but at 6000 miles and my driving conditions, dino is all I need to use with it.
I think it may depend on what you are looking for from the synthetic. If it is for better protection due to severe driving conditions, extreme temps or heavy engine loads, you probably are better off with what you have. If it is just extended drain you want, they say you can go 10000 mile OCIs with just dino oil using LC20. Few people go that far with any oil, including synthetic. If you do not plan on going that far and generally change oil around the 5000-7500 mile mark and still want to use LC20, save your money on the synthetic and just use dino with it. Don't forget to get a UOA now and then just to make sure.