It wouldn't work, for the simple reason that the styles of the two are so different. Star Trek characters (Starfleet anyway) are rather po-faced, whereas the Doctor comes across as a carefree mischief-maker.
Basically the Trek characters would treat him like Q, Trelane, The Outrageous Okona, or Harry Mudd, and they would be like any number of the various blustering space ship crews the Doctor has met over the years..
At first they would - and then he'd team up with them and save the day and they'd realise they're wrong.
I'm not even so sure they would react this way in this particular case, at least not all of them. Sure, the Doctor would kind of cause chaos on the Enterprise (well, maybe Nine less so than Ten) but Archer was pretty easy and in my opinion would be more amused by the Doctor's antics than anything. Reed would probably be angry at first but it would be a nice way for him to learn that there are other ways than his rigid self-discipline.
Well, I'm one of the few people who really liked ENT, so it would have been great. Actually, it would have been great for the clash of cultures behind the scenes alone.
"Mama always told me not to look into the eyes of the sun. But Mama, that's where the fun is."
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