Will my plan work to temporarily power an irrigation pump?
This will be no problem with those wire sizes.
160’ of copper#8 with 28 amps is 2.58%
200’ of aluminum #2 will be 2.13%
So a total voltage drop of 4.71% this is below the NEC recommended 5%.
Note the actual voltage drop at startup will be higher but a standard water pump should be fine with class 10 overloads (standard fuses and breakers are usually ~ class 10 or can handle motor starting currents 3-5x there rating for 10 seconds.
I would bet that pump will be to speed in 1 second with that wire possibly 2 seconds but well below the trip value. So it should work.