Is there a way to do a sysex dump of the entire state of either a BK9 OS 1.06 or BK7m OS 1.07 to insert into a DAW?

I don't believe so. But even if you could... my understanding is that sys ex is very machine-specific. So I don't believe a DAW could make sense of it.

If your goal is to "set" the BK-7m to a certain state, what you might do is to save that state into a registration ("User Program.") It IS possible to recall a specific registration by sending the BK a MIDI message. My $.02.

Thanks for the observations and direction, Ted.
Since I only want my machines set up logged, the DAW would remember the sysex data from my BK and anything else plugged in would ignore it. Hence the "exclusive" in "sysex".
And because it is just data, the DAW would happily record it, even if it did not know what it meant. But, as you say, this does not seem possible.
I shall explore better the section in the manual on "User Programs" and try to do as you advise.

There is no total ‘dump’ of the BK’s full state. Ted’s correct that storing a Performance and having the DAW send a PC/00/32 code to recall it is as close as you’re going to get.

If bookkeeping is what you are shooting for, so all data from a project resides in the same computer folder, you can make the Performance in its own Performance List, then remove the stick from the BK, insert into the computer and copy the UPS file of the List to your computer project file. That will work fine if you only use ROM styles. If you use a User style, you’ll need to copy that into your project folder too.

But that will get you a total backup that no matter how long it is until you revisit the project, all the data you need will be in the one place.

There ARE a few things that unfortunately you can’t back up no matter what, though. These include if you use any custom MIDI Sets, and any Global and Default settings. These aren’t able to be backed up at all (the EA-7 can, though), so if any of this stuff is custom and needed, you had better make a text file of the settings (or take photos of the screens) and store those in your project folder as well.

One last warning... once you create a Performance to be called up by MIDI, do not insert another Performance in the List for any reason! MIDI only calls up the slot number (0-999) not the actual Performance. So if you insert a Performance earlier in the list (Move Performance) the MIDI PC# will call up the wrong Performance! Make sure all new entries get added to the bottom of the List...

Hope this helps.