I'm just trying to sort out the info we get in "The Silver Key." In it, Randolph Carter loses the key to the gate of dreams, which leads him to try writing and occultism and so forth, which is what leads him to Harley Warren. "Statement," then, has to take place after he loses the key of dreams, and as far as I can tell, that has to be after the events of the Dream-Quest
Also note that, while we get something of a recap of his exploits in a couple of other stories to establish that it's the same Carter, we get no such recap in the Dream-Quest
. It starts with his search for the sunset city, and ends with being tricked by Nyarlathotep.
The disconnect between the Carter of the Dream-Quest
and the Carter of "Statement" can perhaps by explained by the dream nature of his prior adventures (and also the fact that when he wrote "Statement," Lovecraft probably didn't know he was going to build on this character later). Sure, he went toe-to-toe with the Crawling Chaos, but after all, that was just a dream. He wouldn't think muich of it when he was awake. The waking Carter and the dream-Carter are practically two different people. Real-world Carter is meek, somewhat cowardly, and easily bullied. Dream-Carter is General freaking Patton by contrast. Perhaps that's why the Dreamlands are so important to both forms ("avatars?" "masks?") of Carter. It's the only place where he can be the adventurer he wants to be.
Then there's the whole "Silver Key" time loop to complilcate things, but I haven't had my coffee yet.