I bought a new bike a couple of months ago. Recently I noticed that the bottom bracket is getting loose. It is regularly unscrewing itself, and while I can screw it back in partially manually, I cannot generate enough torque to get in back in there all the way in. What tool should I use to do this? I have found some 8-notch bottom bracket tools, but they would require removing the cranks, and I'm not sure that they're the correct thing anyway.
It looks like what you have is a cartridge bottom bracket with a left hand cup with both Octalink style 8 notches (even though the axle is clearly square taper) and the more common 20 splines
Park tool BBT-18 (Octalink), BBT-22/32 (splines) with work, as will the dual purpose Lezyne tool you linked to.
Yes, you will have to remove the crank to properly tighten the cup. You probably want to do the drive side as well. To get the cranks off you need a crank puller (such as Park CWP-7). To tighten the bottom bracket and re-install the cranks you need a fairly hefty torque wrench as it's critical the cups and crank bolts are tightened to the correct torque.
For infrequent jobs where special tools are needed it's often easier to have a good local bike repair shop do the work.
If the cup is metal (or even plastic, though greater care must be used to prevent damage), you could try using a pair of plumber's pliers --aka Channel locks--which are tongue and groove, slip joint pliers. You can move the jaws apart to accommodate a wide range of diameters and apply a great deal of torque as well as grip on what you're attempting to turn. These types of pliers are a fairly common household tool. Care must be used to prevent marring the cup with tool marks from the pliers' jaws. An old leather belt or piece of rubber from an old tire or tube, even layers of cloth rag between the jaws and BB will prevent most marks. Selecting the appropriate jaw width that allows the handles to be close together so your hand grip is nearly closed and near max gripping power is key. You don't want the pliers to slip on the BB which will increase the chances of marring the BB's surface. This may seem overly complex or tricky, but it's not. If the channel locks' handle is 10-12 inches long (as even a small pliers of this type can open their jaws wide enough to accommodate the 1 1/4" (32 mm) or so outside diameter of a BB cup), more than enough torque can be applied. The main, cartridge bearing right side takes around 50 N-m of torque, the left cup is to be "snugged" tight after the right is torqued. The torque required at the left is much less. The system will be bound--the cranks won't move--if the left cup is tightened too much. This is my reasoning for using this unconventional technique with a commonly found household tool.