Well, I have suspected iCloud so that is a possibility. Just to point out though NEITHER my MBP or my iPad runs iCloud, disabled on both.
This afternoon I got a message from my sister with absolutely no problem... I even got the "ding" sound when it came in. So it is an intermittent problem for sure.
My messages get archived in my Home/Library/Messages
I don't believe they sync to the iPad... I've never looked for them and that part doesn't matter to me anyway since my sister and I generally chat when I'm on the MBP anyway.
And you are right... it isn't iMessages, it's just Messages... this is how well Apple has me trained! I'm using to everything being "iEverything!"
I wouldn't say my OS's are ancient... the MBP runs El Capitan and the iPad runs 11.4.1. I realize I'm probably looking at upping the MBP to High Sierra. I downloaded and stored the installer, unless I can see something to convince me to go straight to Mojave. Anyway I left myself that option. As for the iPad OS... not until I have to... right now with 11.4.1 I can still do everything via iTunes including syncing Apps as I am able to run the iTunes 12.6.4 they released for that purpose.
Unfortunately upgrades and such for the MBP are on hold for awhile because there are some things that fall under the heading of "life happens" and the risk of a messed up computer is not an option at the moment. Right now I've got as they say "the devil I know."
Just for the sake of trying it, I'm going to move the iPad a bit further away from MBP, instead of it sitting right next to it I can move it over to the speakers that it works with, I can charge it there and everything. It seems odd that messages do come in better on the MBP if the iPad has messages turned off and is not plugged into a USB port on the MBP. Worth a try if it's more consistent.