The bird would be fine, due to the way parallel circuits work (Very little power travels through the more resistant one of two parallel circuits.)
The 'drum like cover' is called a radome, and it's primary purpose is to reduce the wind loading of the microwave dish. It also serves to protect the usually sensitive feedhorn inside the dish, keep out bird nests, ice, water, etc...
The common cause of bird deaths around radio towers is collision, not RF energy. In Florida it is very common to see osprey nests build on/in/around towers. Microwave dishes are a common location because they provide a large horizontal surface to support the nest. See Wikipedia article 'Towerkill'
The physics of electricity propagation in a powerline circuit is fundamentally the same as the propagation of FM radio waves, or even the beam from a flashlight. All of these examples involve electromagnetic energy propagating at the speed of light. So why do we need wires for powerlines, but not for propagating radio signals or light beams?
The principle is that light (in a vacuum at least) travels at a constant but non-infinite velocity, c. Hence, the electrical wiggle received at an observer's location now i.e; (x,t), has been caused by some earlier wiggle conducted by the source (x',t') such that the t-t'=(x-x')/c. We should expect that the effect of this 'time delay' is more profound if the source is wiggling faster in time.
A more specific way to state this is that the electromagnetic power radiated into free space by a dipole increases as the square of the dipole moment and the fourth power of the frequency. Now compare a powerline (60 cycles/sec), with FM radio (100 Million cycles/sec), and with the light from a flashlight (500 Trillion cycles/sec). That's why even atomic dipoles can produce intense visible light, while it would take a very very large dipole to radiate a similar intensity at 60 cycles.
Feynman Lectures Vol.II is absolutely the best reference to learn this stuff.
That is not to say that a bird needs to make contact with two wires on a utility line, in order to suffer harm. If the line voltage is high enough (e.g; 110,000 Volts, as in high-tension power transmission), an electrical corona would form around the line from electrostatic effects. The corona is actually ionized air indicating the high electrostatic field strengths in the region, it can emit a bluish glow and growl at 60 cycles. Birds can no doubt sense this corona & stay away.
So can humans, it's a crackling sound.
How do you think the human standing right next to the tower is not getting fried?
Humans climb live radio towers routinely.
The tissue absorption of RF energy at common AM station frequencies(below 2mhz) is negligible. Maximum absorption happens around the frequencies commonly used by FM stations (about 100mhz)
I have climbed near an live FM antenna. You can feel the heat(well, radiation, converted to heat by your body tissue) from the antenna, it feels like a heat lamp. Your eyes are the most sensitive parts as they have the least ability to dissipate excess heat. The old rule-of-thumb is, if your eyes are watering, you're too close to the antenna. But like most things 'the way they use to do it' -- by that point damage is already being done. Cataracts are one of the first signs of RF induced tissue damage.
It would take many thousands watts, transmitted at frequencies well above common AM frequencies to cause any sort of instant damage. I have never seen a field of dead birds near even major FM stations broadcasting with 50kw+ power.
I made a point of swishing the rotor side-to-side a couple of degrees after I figured out where the dead sparrows came from. That caused them to fly off and once the transmitter was on they'd stay away from the antenna.