As far as I know, Hart's Battery (2nd organization) was never referred to as the Dallas Artillery. For all intents and purposes, it was a new company. Its connection with Dallas and Polk County had been severed when the battery was disbanded by Van Dorn. Also, by this time, the standard practice was to refer to batteries by the name of their commanders, rather than their local names.
The only references I can recall as to the loss of Hart's Battery colors are from Union reports. Van Dorn didn't mention it in his charges against Hart. Again, there is the question of how the Union Army would associate one of the two Dallas Artillery companies with Hart's Battery. I can only speculate that perhaps the flag was recovered from among the abandoned equipment and papers, and thus a connection was made, or perhaps the connection came from prisoners of war. That assumes, however, that Hart's Battery had indeed lost its flag -- something not corroborated by any Confederate accounts that I am aware of.
I don't know anything about the design of Hart's Battery flag, other than what is posted on the Pea Ridge NMP website.