2000 3.0 trans shudder in reverse

2000, 135,000 miles on this car with the 3.0 L. Here is the problem that has come up. I have to back up about 20 ft. in the morning before putting it in forward to exit the driveway. When I put on the brakes and stop, there is a loud knocking/thumping. As soon as I shift out of reverse the noise stops. All of the forward gears are fine (so far). Appears to only happen when cold and first "back up". No SES light. Fluid looks and smells good. Any thoughts? I hear these are prone to failure and wondering if there are any parts/valves that a back yard mech might be able to fix.
Thanks for any feedback,
Jim

So the knocking/thumping continues as long as you are in reverse with the brake applied? If you release the brake and continue in reverse does the noise go away? Can you feel the thumping in the chassis of the car?

Are you certain what you're hearing is not a misfire of the engine? Maybe you have a split in the intake air tube along with a worn engine mount. When you stop in reverse, the engine rocks from the worn mount and causes the split in the intake air tube to open causing a vacuum leak and subsequent misfire.

-Rod