For cool season (TTTF) on my own lawn, I tested 0.4 oz actual iron and 0.8 oz actual iron per 1,000 with COC surfactant (along with a control and a N test).
The test plot with 0.4 oz showed some green up, but turned out to have adequate iron (no real marked response). The test plot at double that, 0.8 oz, showed similar (lack of marked) green up and some blackening. (The N test plot showed a marked response indicating N deficiency).
Some university testing has shown differences in iron blackening depending on temp (more blackening at higher temp).
The upshot is you may need to do some testing in a back corner of your lawn to see how it reacts. Warm season grasses are said to be more tolerant of a lot of iron but my sense is every site has its own limit.