Will my solder joints short out against the aluminium channel?

The short answer is you're probably fine either way, but a bit of tape costs nothing so just do it and stop thinking about it.

Here's the nuance. Anodised aluminium has an oxide layer that's acutally a decent insulator, so a wire brushing against the face of the channel usally does nothing. The catch is the cut ends. Anywhere the extrusion has been sawn, you've got raw bare metal, and that absolutely will short a pad if it bridges 5V to ground or data. End caps and joins tend to be exactly where those cut edges are, which is also where your corner joints are sitting. So the risk isn't really the channel floor, it's the cut ends and any burrs from the saw.

On your two options, tape under the joints is the way to go. Kapton is what most of us use because it's thin and heat resistant, but a strip of good electrical tape works too. It's not overkill, it's standard practice for tight profiles. I'd lay it along the base under each corner joint before you solder rather than trying to squeeze it in after.

As for the diffuser ridges pressing on the joints, that's the part I'd worry about more than the aluminium. Solder joints don't love being under constant pressure, and a rigid diffuser clipping down on a lumpy corner can work the joint loose over months. Two things help: keep the wires short and dressed flat so nothing stands proud, and put a dab of hot glue or silicone under the strip near each corner to hold it down so the diffuser presses on the strip, not on the joint itself. Heatshrink over the corner connections also stops anything snagging on the ridges.

I wouldn't buy wider channel at this stage. 12mm in a 12mm profile is a normal fit, plenty of kitchen runs are done exactly like this. Do the tape, sort the wire routing at the corners, and you'll be fine.

I was thinking more about how much heat the strip might put out since it is tucked in so tight. If all the power is trying to run thru those small copper pads and wires, will the whole thing get too warm over time? It feels like even if the wiring is fine, if the strip gets hot enough, that could mess up the lights. Does packing it so tightly affect how fast the LEDs run hot?

I did a run like this last year and my main worry wasn't the short but the heat build up down the whole run. When you cram the wiring into that small space, the power cable itself gets warm, which can affect the longevity of the strip, especally if you're pushing full brightness constantly. I ended up routing the main power lead alongside the channel but used a couple of foam spacers to keep them separated. It gave the wires room to shed heat instead of just baking in the enclosure. That made sure the whole install stayed stable.

Update: went with the tape as suggested, laid a strip along the base under both corners and slid everything back togather. Nice bit of insurance for the price of nothing. Also took the point about heat on board, so I'm running the lights at a lower level rather than flat out, and after a couple of evenings the joints stay cool. Diffuser clips on fine too, no pressure on the solder.