I hate to be the bearer of bad news, but you're probably not going to be able to carry that particular rifle reenacting. First of all, there were very few civilian arms that made their way into the army after the initial "yippee let's all go to war like our ancestors did" enthusiasm wore off. Civilian guns are in a plethora of calibers, meaning that a unit will have ammunition resupply issues, so they were sometimes rebored to a standard caliber. This was expensive and time consuming and regular arms became available more quickly. 2) Civilian arms were hard to repair, given that they used non-standard parts. So once your mainspring broke, you were not using that gun again. 3) Civilian arms did not mount a bayonet, making them inadequate for military purposes. It may seem silly to us, but the ability to put a pointy thing on the end of the gun was a legitimate military concern when procuring arms. 4) That particular gun has a synthetic stock, which is in no way authentic to the 1860s time period. So, in a word, it's wrong. 5) It has been standard in reenacting for at least 40 years that Hawken or plains style longarms are not acceptable for use on the field at the majority of reenacting events. If your unit won't stop you from carrying it, the event staff will, and you will make your unit look bad. 6) That old "they used any old gun that was lying around" saw is a myth. Disregard it. The South never had serious problems procuring arms. Caps, lead, decent powder, yep - but if nothing else, the Federal government was a prime supplier of functional infantry arms. 7) Carry the rifle your unit recommends. The Enfield is a standard gun in reenacting. 8) Don't show up as a walk-on participant with a hunting gun. You will not be allowed to field and may simply be shown the door, an unhappy result all around.
I can see this is your first post, so I'm letting you down easy here. Just a word of warning: Your question: "I have this non-standard reproduction hunting gun; can I reenact with it?" has been used to troll Civil War reenacting fora for 20 years. I have answered you question assuming that you wanted a real, and compassionate answer. However, you would do well to surf older threads for a while and get an idea of what's being discussed, or has been discussed in the past. If you come back with "can I wear blue jeans?" or "can I wear this leather coat I found in SalVal?" you may not get as polite a response.