Well, the saga continues. After a couple of hours on the phone with Brother yesterday the new printer and Mojave will not communicate regularly. A search shows this to be an issue with several printer brands even though the Brother drivers are supposed to be 10.14 compatible.
The case was escalated and I'm supposed to receive a call from a Brother guru further up the chain this morning. If that doesn't work, back it goes.
I will certainly give credit to the support person, he was patient and we tried everything. Twice. The only thing that he didn't suggest and I tried on my own was getting into the Firewall settings to make sure the printer application could access the computer. Still no luck.
The printer and wireless router are set the same. This has been triple checked because I have a cable company Ubee router as well as my own TP Link and each has 2.4 and 5G settings, thus giving me 4 options. It's my understanding that the 5G is not to be used with most printers. Brother's website shows a known issue with the Ubee losing touch with the printer so I avoid that one. I have attempted to set up AirPrint but it fails every time. I can occasionally print a test page from the printers and scanners page. But to try a Pages document or to use Brother's iPrint & Scan app shows no printer available.
So for now all printer drivers and files for any printer that's ever been used are uninstalled and I await Brother's call to see if we can iron this out.