I asked a Laramie County Master Gardener for some help with your question and here is her response.

"If you have 'Cheyenne Spirit,' he/she might be growing an Echinacea (in this case Echinacea x hybrida) rather than a Rudbeckia. Most of the Echinacea species are tap-rooted (with E. purpurea being an exception) and don't divide well. 'Cheyenne Spirit' might have a fibrous root system, though, and that might enable the gardener to get a viable piece to share.

 

With respect to growing from seeds, commercial seeds of the original 'Cheyenne Spirit' cross are available. The few times people have sent me their saved seeds, though, the seeds weren't viable. This does not mean the plants do not produce viable seeds. It's only an indication that the particular seed savers didn't know how to collect/distinguish viable seeds. I use what I call the "thumbnail test" to determine which seeds are more likely to be full and mature. (Use just enough pressure to feel if there's a firm, mature seed inside. The flat/empty/mushy-feeling "seeds" aren't viable.) Here's a link to a post from a couple years ago: https://garden.org/thread/view_post/1836718/  (If you read the rest of the thread, disregard the information about which calendula seeds are viable. We debunked the information in the thread in a separate discussion.)

 With respect to seeds of 'Cheyenne Spirit,' it would be a very fun experiment to collect viable seeds (if indeed the cultivar produces viable seeds) and grow them out to see if the progeny resemble the parents.

 Regarding whether a fall or spring division attempt is more likely to succeed, in your experience, do Echinacea species tend to establish equally well in either season? I put out some E. angustifolia seedlings last spring and last fall, and both groups made it through the winter."

From my perspective, starting from seed may be the best option rather than trying to dig up a happy plant and have an unhappy one later. 

Hope this helps

Thank you so much for this information!  I will probably try the seed idea and maybe also a little digging.  Your reply raised another (minor) question for me.  I know that Rudbeckia and Echinachea are related plants and often see them interchanged on plant tags.  I have the Cheyenne Spirit (which was labeled Rudbeckia), also a purple coneflower (classis echinachea), and a new yellow black-eyed susan (also labeled Rudbeckia on the nursery tag). What is the difference, or is this just a genus/species type question.  Just trying to learn!

Echinacea purpurea aka Coneflower and Rudbeckia flugida aka Black-Eyed Susan are both in the same  Asteraceae family (think sunflowers).
Echinacea is in the tribe of Helianthodae and has a tap root (makes it hard to divide). However; Rudbeckia has a stoloniferous root system (it likes to creep and spread via roots) making it easy to divide. 

hope this helps

Thank you!