Need help with a skin rash

Petey developed a skin rash a few weeks ago. I suspect it may be from a Rhus lanceolata
prairie sumac stand of trees that have made themselves a home on the fence line.
They don't bother our other dogs,so I am not sure.
Has anyone experienced problems from this tree? Petey likes to lay in there shade when he is outdoors.
I have been keeping him inside as much as possible, but the rash remains, but not as bad.
I would like any help offered.
This is the only plant outside that makes me wonder since it says in plantfiles that it can cause a rash.
I hope it isn't allergies as I have another dog that had allergies when I acquired her and I hate to see a dog having problems that can't be easily helped.
I am pretty sure this is the type of Sumac that it is by the pictures. It doesn't ahve a fuzzy trunk like the Staghorn pics I saw and I am almost positive it is not poison sumac as I am not bothered by it at all.It grows in a very dry ground also and the poison sumac description says it is a wet area tree, shrub.