Not only is the top half of Vermont insanely difficult, daylight hours are getting short and that will cut into your mileage. 1 MPH is a typical pace for the top half of the LT. To E2E the LT, your looking at 3-4 weeks.
Also keep in mind that outside the National Forest land, camping is restricted to designated sites only and tenting at many of these sites is limited or non-existent. In general, the LT is not very tent friendly. I typically carry just a bivy sack for emergencies. It's easier to find a place just big enough to lay down on, then it is to set up a tent. Just hope it don't rain that night. Collage groups are at every site in early September. Typically they find someplace near the shelter to set up a big tarp they all sleep under, which farther complicates things
I would not recommend bringing a dog. You will have to carry him up or down some places which you need both hands to hold onto rocks and trees.
How's the dog at climbing aluminum ladders?
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Or how about getting up this? Yes, that's the trail!
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There are some really scary parts of the trail on Camels Hump and Mt Mansfield too.
Other then that, September is a great time on the LT. It will get really nippy towards the end of September too. Frost is very likely, maybe even snow if you go NOBO.