He has an angry wrenlike vigilance, a greyhound's gentle tautness;
Lowell is back to using the human pronoun "he." But he quickly departs from human descriptors when he brings the wren in.
In these two lines Lowell describes the Colonel using two different animals: a wren (type of bird), and a greyhound (a dog).
In line 33 he describes him having an "angry […] vigilance." Vigilance, btw, means keeping careful look out for something dangerous, which would certainly be part of a colonel's job.
In line 34, instead of angry, he's described as having a "gentle tautness." This isn't exactly the opposite of the previous line, but gentle and angry don't usually work together. So the Colonel seems taut (like a bow), but in a gentle, maybe natural, way.
This isn't the first time we've seen seeming contradictions used to describe the same thing. Remember snow in the Sahara in the first stanza? It looks like this is becoming a stylistic choice we can expect from Lowell in this poem.
he seems to wince at pleasure, and suffocate for privacy.
Lowell is reading into what he sees, and is personifying the statue, or giving human feelings to it.
"Wince at pleasure" is a strange phrase. You'd think pleasure would be something you'd want to welcome, but maybe this vigilant colonel is so used to bracing himself for danger, that he'd be completely unprepared for anything good to come his way.
"Suffocate for privacy" is equally strange. Maybe it means that being on display in a public place all the time is eventually a little uncomfortable. I mean, sometimes you just want to kick it by yourself on the couch in some sweats. The Colonel here isn't getting that kind of alone time.
Let's track the progress of the statue's description: Lowell went from describing the statue as a mere object (using "Its"), to comparing it to animals ("wrenlike" and "greyhound"), to thinking about its emotional status (clearly a more human example). So the evolution of the statue's description brings it slowly more to life as the poem goes on.