Missoula to Astoria via L&C is about 690 miles. If you actually want to touch the Pacific Ocean, you'll need to go another 15 to 20 miles. (Assuming you stop in Yorktown, however, you won't technically touch the Atlantic Ocean either.)
Missoula to Florence via TA is about 840 miles. (It's 150 miles more to Astoria, or 130 miles more to Seaside, instead of Florence.)
Depending on how far you go in a day, L&C would save you two, maybe three, days. You could save just as much time by averaging 2 miles farther each day as you cross.
If you get in a hurry, there are twenty ways to save 10 miles here, 20 miles there, as you cross the country by deviating from the ACA route, but I don't recommend any of them. One such way is to cut off that little side trip into Missoula, but I don't recommend that either unless you have to because of time.
I've done the TA but not the L&C, so I cannot compare the routes. It's probably six of one, half dozen of the other. Fun either way. But you don't want to be in too much of a rush.