Hi Brian,
You should be able to use ArcPy to do this. In the documentation you linked Esri has a code snippet:
import arcpy, os
pdfPath = r"C:\Project\ParcelAtlasMapBook.pdf"
if os.path.exists(pdfPath):
os.remove(pdfPath)
pdfDoc = arcpy.mapping.PDFDocumentCreate(pdfPath)
pdfDoc.appendPages(r"C:\Project\Title.pdf")
pdfDoc.appendPages(r"C:\Project\ParcelAtlas.pdf")
pdfDoc.appendPages(r"C:\Project\ContactInfo.pdf")
pdfDoc.saveAndClose()
del pdfDoc
Line 9 will create a new PDF (folder\WHBFD364.pdf would be the path passed here), then use the appendPages function in subsequent lines to append the other PDF's into the newly created one. Make sure to append them in the order you would like them to appear in the new PDF. Then, save and close the new PDF in line 15.
Once the appends are done, you could remove the old PDF files using Delete—Data Management toolbox | ArcGIS Desktop or using the os python package's remove method like in line 6.
If the ArcPy method doesn't work, there is also the PyPDF2 Python package. That package has a PdfFileMerger class that can be used to merge multiple PDF's into one. Here is a stack overflow post with some examples: python - Merge PDF files - Stack Overflow.
Hope this helps!