It gets worse with more force, like when climbing.
I will admit I was the last one to put the cranks on and I didn't have a torque wrench, but I'm pretty certain I got them on securely. At least there is no lateral play in the cranks.
I'm thinking that leaves either the pedal bearings or the bottom bracket as the cause? Does anyone have any other ideas? Is there a good way to diagnose whether the pedals are the cause without having to buy new ones?
Crankset is a Shimano 105, two-piece compression slotted, non self extracting
Bottom bracket is Shimano SM-BBR60, about 5 years old.
Pedals were replaced due to pitted bearings about 3 years ago.
Your noise is pretty loud and constant. It suggests something is worn out. It's possible there's some interface that needs cleaning and greasing, but with Shimano road BBs, that usually won't produce a noise so constant and obnoxious. This is what I would do:
Pull the cranks and wipe the grit off everything, and also wipe down the contact surfaces of the spindle.
Check that the BB isn't overtly worn/gritty feeling. 5 years of riding is an ample lifespan for Shimano external threaded BBs, and they're inexpensive to replace.
BB cups torqued.
Chainring bolts torqued (this isn't it if you're sure it's only from left side pressure).
Light grease cooating on the the black BB top hats in the spindle contact area. Shove the spindle through. Don't grease the spindle itself because it will squeegee into a big mess on the drive side. Grease the inside the of the splines on the left. Install the left side, adjust so the play is just gone, torque down both bolts to 14Nm in a row. (Or, sans torque wrench, make them both as tight as you would ever reasonably tighten something with a 5mm L-wrench.)
If that doesn't do it, test with different pedals. Alternately, sometimes you can get this right off the bat by applying pressure to the left crank at the end of the crank arm, getting no noise, and then applying similar force to the pedal and getting obvious noise.