Disney is very firearms unfriendly but the issue here is while, if stopped, they will tell you firearms are not allowed at Disney Springs, it's not posted on property nor is it anywhere (except at the AMC theater which is AMC's policy) in any documentation that a guest has access to.
They are very upfront about it if asked about the parks and resorts, but seem to side step that question when asked about DS.
At DS, what they can do is ask that you return to your vehicle and secure the firearm before being allowed to return to DS proper or ask you to leave if you refuse. At that point, you would be informed that refusal to comply will result in being trespassed. FL law allows for a permitted person to be allowed to secure their firearm in a vehicle if they are at a place where concealed carry is not allowed by the property owner.
If a dog hits on you and you did secure your firearm properly, all they can say is thank you, have a magical day.