Where is the mistake in this air density calculation?
The mistake is subtracting the ISA temperature at sea level from the ISA+18°F temperature at altitude in step two. If your atmosphere is heated up, it needs to be hotter at sea level as well, or your lapse rate in the first step and the denominator is wrong.

First calculate density for ISA conditions. This gives 0.7708 kg/m³.

Then used the ideal gas law to see how a 10 K temperature increase will reduce density at constant pressure. The density ratio is inversely proportional to the temperature ratio, so the result is 0.7421 kg/m³.

When checking your result, input only 10 K into the calculator, not 18 K !!!