You're right, this phrase is just dripping with unacceptable themes.
First, the word "little" is offensive to vertically challenged persons. The OP should be more sensitive to their feelings.
Second, the word "cowboy" is really dog-whistle speak for "cowboys and Indians" which is offensive to all native Americans. Also, this is gender-baiting since it is not inclusive for young girls. Finally, PETA does not condone anything supportive of ranching and the use of animals such as cows.
Finally, the word "room" is particularly offensive to homeless persons. After all they have no room of their own.
Around Tombstone, cowboy referred to an unsavory type. (1800’s, not that I’m that old)