This sounds kind of familiar. I've fished in my fuel tank too. And with about the same results. Ended up removing it and holding it upside down. But maybe that isn't necessary here.
Lets back up a step or two. I kind of doubt that the cap or screen or whatever that fell into the fuel tank is what is making the tractor run poorly. My guess is that whatever was being screened out of the fuel got dropped into the tank and passed from there down into the fuel filter where it has plugged that filter up. That's common enough, and a plugged fuel filter exactly fits what you said about what is happening as it runs. The fuel filter is plugged up and the motor is starving for fuel.
So I'm afraid that even if you do get the elusive screen out of the fuel tank you won't accomplish anything except satisfaction - you are still going to need to clean out the fuel filter. So I'd leave the tank alone for now and concentrate on the fuel filter. I'll also bet that cap & screen can flop around in the tank for a long time without damage to anything.
Every tractor has a fuel filter - often the filter lives in a clear plastic housing so you can see dirt and water. To find the filter, follow the rubber hose coming out the bottom of the fuel tank, and that leads right to the fuel filter. There is often a little handle on the fuel filter housing that can be used for turning the fuel flow on & off when you stop the tractor, or to stop the flow of fuel out of the tank while you work on the filter.
And there is a little finger-push valve on top of the housing for burping the air out of the housing after it has been cleaned out & reassembled. This is easy stuff to do even if it is a bit bit messy it is less so than mucking around in the fuel tank. You'll want a plastic container, a brush for cleaning the filter and housing, and a few pairs of pliers plus maybe some waterpump pliers,..... & a screwdriver or two. That ought to do it. The filter housing us almost always deliberately made to be simple, accessible, and easy to deal with by owners with simple hand tools. It's made that way because all over the world, dirty contaminated diesel fuel is common, and cleaning out the fuel filter is a common job.
Enjoy!
rScotty