What causes the tinny sound in the exhaust?

The "tiny" sound is probably caused by the exhaust ringing from the resonance of the exhaust pulse wave coupled with thin wall tubing.

Check the heatsheild above the cat, those things are notorious for metallic rattles.

Thin wall headers and/or thin wall exhaust. It took me a little bit to get used to after I installed my headers.

Is there any way of eliminating that sound without changing the exhaust?

Put your stock manifolds back on and it will go away.