I think the original post is causing a bit of confusion due to terminology.
You can in fact have a constant speed prop with a blended airfoil. As is the current crop of Hartzel constant-speed versions found on many RVs. The blended airfoil prop blade is not limited to fixed-pitch propellers.
So propellers. We have fixed pitch. Ground adjustable. Variable Pitch (no governor, the pilot is the governor). And Constant Speed (the modern de facto changeable prop blade angle design). We even have what has been coined "almost constant speed" where a flexible fixed pitch prop changes shape due flexing due to loading. Bernie Warnke kind of pioneered this type of prop in the 70's and 80's and they are still around today. As Paul Dye pointed out these modes of operation have little to do with the propeller geometric planform, like "blended airfoil", "scimitar" or legacy straight and square. They are two different things. The former being mode of operation and the latter being a description of geometry based on aerodynamic objectives and aerodynamic design improvements.
There isn't any way a fixed pitch prop, of any planform profile, could provide the climb and cruise mission profile as you have specified in your requirements. There isn't anything wrong with the simplicity and light weight of well-designed fixed pitch propellers but it sounds like you are aiming for all the performance possible from an RV. One of the most endearing qualities of Van's aircraft is the spread between low-speed performance and top end cruise performance so it makes sense for you to strive for both. RVs are docile and safe with good pattern behaviour without surprises. So taking that operational range spread into account, in order to meet your specified mission profile you will definitely want a constant speed propeller.
It is rare to run into a "variable pitch" propeller as they were replaced by "constant speed" props. The variable pitch propeller depended on the pilot to constantly move the propeller control to keep the pitch desired without over speeding the engine. I flew a Stinson 108 with an O-435 with constant speed a few times and the prop control was almost a primary flight control. So a variable pitch prop is analogous to driving a manual transmission automobile where the driver shifts gears based on speed and power requirement. A constant-speed prop is a step up and is more analogous to an automatic transmission vehicle where the pilot sets whatever RPM he wants and the governor takes over the tedious task of automatically adjusting the propeller blades to maintain that RPM. The governor vastly reduces the pilot workload and can prevent engine damage due to overspeed in a dive. At the same time pushing the constant speed prop control can help to add drag in the pattern while configuring the aircraft for a proper go around. There is a reason why most high performance piston aircraft go constant-speed.
So that's the answer if you are asking about constant speed versus fixed pitch. If you want to find out about the geometrical attributes, blade loading and geometric design of a propeller blade irrespective of operating configuration that is a wholly different discussion. There is plenty dynamic physics, aeroelastics and aerodynamics involved with about as much math as empirical data obtained through testing. The most successful propeller designers can get the reputation of practicing a black art as it can appear to the casual observer to be about as much art as science. Suffice it to say they are very passionate, capable, smart and driven to get the most performance possible through lessons learned.