That walk sounds bl**dy cold !
Something I've used for exhaust staining, which works very well if you have a
clear access to the area to be stained, without obstructions such as lumps and bumps, is a ladies eye-liner pencil, or a 'Chinagraph' marking pencil.
Draw a light line representing the staining, and then smudge it with a fingertip, spreading it back along the flow line. Further applications can make the staining heavier if required.
These pencils are available in different colours, with the black and various browns being useful. However, depending on aircraft type, the fuel used, and the engine settings etc, exhaust staining could range from whitish grey, through blue-grey, to light brown and almost pink !
Here's an example of the 'pencil' staining.