You are wise to want to keep heat trapping dust from building up.
And you are wise to want to avoid removing the cooler.
As indicated, using compressed air is best. But there are a couple precautions. First, do it outside. No need to blast that heat-trapping dust, dander, microscopic critters that eat that dander, and the microscopic "stuff" those critters "excrete"
back into your room, computer, nose or lungs.
Use a wooden glue/Popsicle stick to hold the blades stationary while blasting. You want to avoid over-spinning the fan as that can damage the bearings.
You can find cans of dusting gas at most home improvement and discount stores like Walmart. But it is important to understand those cans of dusting gas do NOT contain “air”. Instead, they contain difluoroethane or a similar gas. These gases are typically used as refrigerants and are extremely hazardous if inhaled. In extreme cases, it can damage the central nervous system, or even be fatal. So use in a well ventilated area or better yet, outside. Note in some jurisdictions, these are controlled substances and sold in a controlled manner, similar to how cans of spray paint are sold to [hopefully] dissuade abuse through “
huffing”.
Also, make sure you hold the can level or else it will spew super cold, liquid gas on your electronics.
The best solution is to use an air compressor. Just make sure to use it with a
inline moisture and particulate filter when cleaning electronics (or air-brush painting birthday cakes or faces). This is how I clean electronics. But again, keep the fans stationary when cleaning. For stubborn crud (like sticky pet hair and dander - especially from cats - and/or smoke residue) a clean, soft, natural bristle (not synthetic due to static) dusting or paint brush can help dislodge it.
Finally, make sure any case you buy in the future has removable, washable air filters. These, as a minimum, cut in half the number of times you need break down the computer to lug outside to blast out the heat-trapping dust. Of course, your situation will vary depending on your environment, number of kids and pets shedding dander and running around stirring up dust, and hours per day your case fans are drawing in dusty air.