Your image of 3872 x 2592 pixels printed to 36" wide would give you about 107 pixels per inch. As you won't be viewing this as close as a 6X4 or 10X8 you probably don't need 300dpi. I would suggest that you send them a full size image (don't worry about the dpi that's reported) and let them do the up-sizing. The 72dpi that is shown is only a flag to tell the printer how large to print the image it has no bearing on the size of the image.
You can check this yourself by saving copies of the same image at 300dpi and 72dpi and you will find they are the same size in disk space and they display on screen the same size. It's only when you 'print with preview' in Photoshop or equivalent you will see how large the printer will make the image. You can override the default size and tell PS to print-to-fit or to any physical size you like.
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