I would like to swap the 26 T for a 22 T, which would get me down to 18 gear inches. But even with the 26 T, the chain starts dragging on the front derailleur 4 gears up from the bottom on the cassette.
You shouldn't be using the smallest four cogs when you're in the smallest chainring anyway. The cross-chainging reduces the life of the equipment, and those gears overlap ones you can get in the middle ring. The same goes for using the largest few cogs in the big ring. Get those ratios in the middle ring to keep a straighter chain line.
According to Trek's website, the new 520 has the M543 crankset. I can't find it on Shimano's website which is not as good as it was before they revamped it a year or two ago; but since the middle ring is smaller than 38, I guess it's not a road triple. Road triples normally have a 74mm BCD for the inner ring which lets you get down to 24 teeth. 5 bolts is standard. If this one is a 4-bolt one for MTBs, you can probably get down to 22 teeth.
If you want to get lower, you could use a cassette with a 34-tooth big cog.
Since my wife is anything but a climber, and we've done some super climbs like 8 miles of 10% on our road tandem, I put a 24T small ring on it and a 13-34 cassette, giving us a low gear of about 19" with 700c wheels. So the crankset is 24-42-52, shifts fine (as well as can be expected for an inexpensive crankset whose chainrings don't have ramps and pins), and never drops the chain. One secret to adjusting the front derailleur to work that well is usually missed: The outside plate of the cage should not be perfectly parallel to the outer ring, but instead, the rear of the cage should be slightly
farther out than the front is.