So I just double checked for you on 4 different configurations that are in front of me at this time. And as far as I can tell, this issue has not resurfaced. I have 2 PCs running Windows 7, both running MSE(Microsoft Security Essentials) and one is running Malwarebytes Corporate, the other is Running the Normal Pro version of Malwarebytes that is the most common version. Neither of these PCs have been running since the start of the Christmas Break, so they have both been off for almost 3 weeks, meaning both Malwarebytes and MSE needed to be updated on both machines.
Neither PCs had any trouble updating MSE while Malwarebytes Ran, which tells me this bug, at least on these configurations, has not returned. I also double checked for definition updates manually, and then watched my firewall to see if it actually reached out to Microsofts server, and it did. So no issues I could find. I also did the exact same test on my other 2 systems, only difference is they are both Windows 10 systems, but otherwise all other variables are the same, one Corporate and One Pro, both had been off for several weeks.
However with that being said, 3 systems are fairly newly built Ryzen 2000 Series Builds, and One system was an Older 4th Gen i7 Built system, so that hardly scratches the surface on testing possible hardware configurations, so I don't doubt that you are experiencing a problem. I would want to investigate your situation further before jumping to conclusions, your issue could be caused by a endless number of possibilities.
But if you are reasonably certain the problem is in fact Malwarebytes, you asked if there was a way you can turn of Malwarebytes, then Yes there is. Just open Malwarebytes up, turn off all protection, then exit the program. Then Malwarebytes will no longer be on, so if that was the problem, you can proceed with your update of MSE. Once MSE is updated, you can open Malwarebytes back up, and turn all protection back on.
I would also make certain you update Malwarebytes all the way too, as having it up to date might fix the problem you are experiencing.
I would also make sure to completely Uninstall all other software virus scanners other than Malwarebytes and MSE.
You also asked if MSE and Malwarebytes are compatible with each other, yes they are. They are the best to security programs to use together, and all you need in most circumstances.
With that being said, Microsoft does not OFFICIALLY support MSE running at the same time as any other anti virus program, but in practice it has ALWAYS played nice with Malwarebytes, except for the one or 2 times bugs were accidently introduced, but both times the bugs were fixed, and they have always gone back to playing nice with each other afterwards!
Anyways, I hope that answers all your questions. Just keep both updated and you should be fine, if you still experience issues, look to make sure there isn't some other anti-virus or software firewall installed that is causing the issues for you!