It will fit fine, don't over think it. IIRC the spacer is just for fine tuning and may not be needed.
On one bike (Lightspeed analogous to your Trek, 2001 model) I think I took a piece of plastic
that covered the bearing opening outside the BB on the L off, on another I fit a narrow spacer
on the axle inside the R crank between the crank and BB to get a better chain line, both used BB similar to
the DA I referenced. The axle slides into the BB from the R side and the L crank clamps onto the
axle splines. If there is any slop left when the L crank is fully seated, you may want to try a spacer.
Last time I did this was 3 and 5 yrs ago so I may be explaining in less than full mechanic mode.
Shimano has all tech docs on line readily available and you can look at www.parktool.com under repair
for more info.