I wonder if there's anything in the BIOS or perhaps some software that I can use to control the fan better?
I've already told you so, you should use SpeedFan to control your fans: have you tried it?
About your BIOS I don't know, check the manual about which options are available: I think that BIOSTAR one should have at least a rather sophisticated CPU FAN management in the PC Health Status section.
Another option is an internal fan controller, like the mCubed T-Balancer, otherwise you may use as many Zalman Fan Mate as you may need.
Moreover, your GD05 has three 120mm intake fans: they run at 1200rpm for a noise baseline of 24dB, way high IMO, providing that you have relatively low power parts.
If you want to run positive pressure, I think that you should dial down them (at no more than 600rpm): you may use either a cable splitter at a fixed voltage one (7-8V I guess), or something like the Akasa one for PWM if you don't want to run them at fixed speed.
Otherwise, if you would run a classic negative pressure case, then you have to disconnect two of them and dial down the third (you have to do some tries to determine which one to left on, probably one or both of the right side
The CPU is a 45w AMD and it almost never goes above the mid 30°s. Do I need the fan on as much as it is now?
Absolutely no: you can easily cool such a CPU with your Scythe fan at about 300rpm. If you have picked the smaller Shuriken, then it's mandatory, as its 10cm slim fan may be really annoying at over 2000rpm.
If you were going to build the best quiet system, what would it be?
The quietest "compact as a Shuttle is"?
BTW, if you can switch its PSU (even with some Pico-style one, if it's enough) and its CPU cooler, you might get some huge improvement even with that Shuttle.