I do a fair amount of airbrush painting for my masks and other objects. Although airbrushing is good for such things as stripes and so on, I don't think it's exactly necessary for a single tone outfit. (Unless you're going for the grey on top, white on the underbelly and limbs look).
If you'd be happy with it, and have the ability to purchase another one if things go awry, I would go ahead with the dye job - although they do make grey fabric dyes. It will last longer as long as the fabric will take the dye (some synthetics have issue with it) and you do wash it throughly afterwards (getting grey all over your skin and costume due to sweat or rain sucks)
To create a 'fur' pattern, my suggestion after you get the grey down would be to take some black acrylic paint (or fabric paint if you'd like) and a 2" disposable brush and do some dry brushing on the fabric with long strokes. Test your technique first on a piece of spare fabric (even stuff isn't what you're going to be painting) EVERY time you reload then wipe down your brush as something it only takes one heavy loaded brush to ruin it all.
Note that again, you should be wearing the costume as much as possible when you do this as the stretch of the fabric and result in the paint 'breaking'. I'll see about doing a test of what I mean this weekend and posting the resulting pics here. Another possibility is using india ink as the 'paint' so that it stains the fabric instead of painting on it but I've never tried drybrushing with it. Maybe I'll do a pattern of that as well - just to have it on record.
After you're done, you can set it with heat by putting the costume onto an ironing board, placing a sheet of broadcloth over the top of it and then using a dry iron at the appropriate heat for your garment, give it a good ironing.
I have found that 30 seconds of going back and forth the length of the ironing board tends to set it well - although some fabric paints I believe suggest longer. If it is a spandex type suit though, you'll want to be careful with the heat as the polymers in the fabric can melt. Note that heat setting of paint isn't always necessary but I find it useful if it is something you plan on washing every event.
Also, and not sure if your outfit comes with it, but adding some faux fur that goes with the colors onto your shoulders helps complete the illusion. Some people also go for wrist and ankle fur as well but with clothing over it, I find it a bit of an unnecessary addition.
