Your oil change intervals are way too short. You cite a 3,000 mile (5,000 km) oil change interval with conventional oil. That interval went out with the Reagan administration. It can be roughly 3 times that long with normal driving. With full synthetic you should be able to go well beyond 10,000 miles (16,000 km).
If you insist on using synthetic I would change oil and filter every 10,000 miles (16,000 km) - to stay within warranty - and rotate tires at the oil change and once in between (i.e. every 5,000 miles). Just my humble opinion.