(1) What we discovered (I think) is that -- once the BCD is rewritten with a letter in it -- every reboot brings the letter back to the partition in Windows too. Partition 6 is not mentioned in your BCD, so a letter on it won't matter. Only your Windows partition (C:) & your Recovery partition are mentioned in your BCD. So -- go ahead & do that DISM. You'll be able to remove the letter afterward.
(2) "and we now have cleared those files from BCD, so those are no longer available to delete in order to force BCD to re-whatever"
You've got nine more useless entries in your BCD. Here is the next one...
Windows Boot Loader
identifier {49275add-50fb-4501-8b3b-7df483700c94}
device unknown <<< Nothing to load -- where is it?
path \Windows\system32\winload.efi
description Windows
osdevice unknown
systemroot \Windows
nx OptIn
It's harmless, but unsightly. It used to load a copy of Windows, or it was in construction to load one. But it's got no device (partition that holds Windows). I can only guess why you've got so many of those. Something sloppy must have been going on. Delete it with...
BCDEdit /Delete {49275add-50fb-4501-8b3b-7df483700c94}
That deletes the entire entry -- all of the lines above. And you'll have eight more entries to go. Careful, only delete the ones that say "device unknown", though.
(3) "Yes, I can male a backup of the drive today. Although, don’t we need only a backup of part 6?"
It locks in the good we've already done with Partition 5 (WinRE.wim) to back it all up. Also, we're going to mess with settings in parts unknown, if we do a ReagentC /SetOSImage regarding Partition 6 (Install.wim). I thought they would be in the BCD. They aren't, but must be somewhere.
I wish it wouldn't take hours for you to make an image, though. If/when Acronis no longer forces a sector by sector, it should be much quicker, just as you say. Anyway, DiscWizard is well under 15 minutes for me -- & I expect that would only triple, if I had a horribly bloated Gateway Win 8 like you.
(4) "BTW, your comment about getting space moved between sectors, I might be able to basically leap frog over partitions by using Partition Wizard."
If we grab sectors from C:partition (shrink it) & leap frog them rightward past Partition 6 (so that we can have a partition 7), what happens is that Partition 6 moves leftward into sectors once held by C:partition.
(5) I found this...
Add a custom tool to the Windows RE boot options menu
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That could be required to establish the actual push button we've been looking for. I think Gateway did something like that to put it into Advanced Startup. And it's gone now because Windows decided to wipe it in 2017 when it overwrote the WinRE.wim that was in Partition 6.