Altitude won't be a major factor - -
Wind will.

You can COUNT ON strong headwinds in west Kansas and eastern Colorado / New Mexico - especially in the fall. The region has one of the strongest prevailing wind patterns - winds out of the southwest. And you can NEVER beat a wind - the wind will only beat you. Best ways to deal - start as early as possible since the wind picks up as the day progresses - with peak winds in the mid to late afternoon. In fact, there is sometimes a light return-flow wind just before and after sunrise. If you know where you are going to stay for the night - riding an hour or two before sunset also avoids the wind - but not as much as the early morning. And if the wind is really bad - just take the day off. It beats going crazy.

I just did 1000 miles in less than three weeks - including 7 big mountain passes.
I'm 51 and a bit overweight - so I'm sure you can do it in your allotted time.
If you ride all day it's hard to do less than 60 miles per day.
St Louis to Taos is about 1000 miles so I would guess 4 weeks - tops.