The camp stove fuel (white gas) leaves no residue so soaking the entire shutter will do no harm n leave the blades nice n clean.. no fussing at all later. If there is still signs of stains n oil on the blades, soak it again till it is all gone, you haven't done a complete job.
Use jars for soaking your shutters so you can swirl it around in the solution. My findings, over the years, has been the soaking method does a pretty thorough job. If the shutter refuses to work after a good soak it may need to be broken down to get the last of the stuborn heavy gunk lodged in the works generally put in by a DIYer, or there may be damaged n worn parts. Use very tiny amts of lube, think more on a molecular scale.
Using a paint brush only smears the old stuff around n foces the oils n grease to lodge itself in other places after the lighter fluid evaporates, not really cleaning it out. Soaking disolves the oil n grease. I like to use a 3 step wash so I have fresh solution as I progress.
Coleman fuel is naptha..
White gas is also sold as other brands like MSR, it is camping stove liquid fuel AKA white gas.