Is it feasible to design a flight computer for a small CubeSat spacecraft destined for lunar orbit that is controlled exclusively by one or more non-radiation-hardened processors? What are the main design considerations to make such an architecture radiation tolerant, without using special radiation hardened components?
What would be the expected risk of single-event upsets and total ionizing dose effects over a one-year time span?
What type of redundancy or voting scheme, if any, would be recommended for the processors?
What shielding, if any, does the larger spacecraft provide? Shielding knocks down the derived environments to those the processor and other susceptible electronics will actually see.
What type of computer is required? Key requirements include operations per second, memory, storage, etc. Different types components and technologies respond to radiation environments differently. Your performance requirements may rule out entire types of technology.
When does your computer need to operate? Is it on during launch, transit through the South Atlantic Anomaly, Van Allen Belts, or other localized areas of high radiation environments? It’s probably an easier mission if it’s only turned on once it reaches Lunar orbit.
Probably others I’m forgetting that I’ll add based on comments.
Now that we've enumerated some complexities, let's blithely brush that away and talk about a potential architecture. You specifically want a processor that's not radiation hardened, but there are a variety of other radiation tolerance approaches you can use.
More shielding. This time around the computer or processor. Aluminum is a good, cheap, and lightweight shield material. Tantalum is another high-performance shield material.
De-rating. Use components below their specified values. This makes them more radiation tolerate. MIL-STD-975M has de-rating advice.
Voting. Have multiple copies of your most susceptible components (probably processors or memory) that vote to determine truth. Voted EEPROMS can be a good way to store a golden master of key information that can be restored if error-detecting memory finds a flipped bit that it cannot repair.
Watchdog timers. Latch-up is a radiation failure mode that can be recovered from by power-cycling. Have a robust analog watch dog timer that isn't radiation susceptible that must be pet periodically by the processor. If the processor hangs the timer resets the power to clear any latches. Of course this requires your mission be robust to having the computer go down for a little bit as things reboot.
If you can't have the computer go down you may want a redundant processor running in a shadow mode and automatic fail-over to the secondary computer.