There is really no reason not to use synthetic in your Briggs. While the Briggs will run for many, many years with neglect, why not pamper it…after all, the cost is so minimal.
If you can get a hold of some Schaeffer’s Neutra, you could safely add about 1 oz of Neutra to your Castrol Syntec to give it a good cleaning. You could safely leave the oil with the Neutra all season if you wanted.
In my Honda OHC mower and my Briggs 3.5 edger motor, I run Amsoil 15w40 Heavy Duty Diesel and Marine oil with the addition of 1 oz. Schaeffer’s #132 Moly E.P. Oil Treatment. In my 3.5 Briggs rototiller motor, I run Schaeffer’s #700 Supreme 7000 15w40 synthetic blend oil.