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Changing a drive letter in Windows won't directly affect Rescue. There's nothing recorded on the partition itself that says, "I should be drive letter X." Windows remembers what drive letters have been manually specified to specific volumes, but that's due to Windows maintaining a memory, not anything stored on the disk. Reflect does bake some information into Rescue Media about drive letter assignments so that if you use Rescue Media on the actual PC where you created it, then often drives will have the same letters you normally see in Windows -- but even that isn't perfect, and you'd have had to perform a full WIM rebuild of your Rescue Media after changing the drive letter for that info to be incorporated into it.
In terms of the issue, try opening PE Explorer in Rescue Media, which is Macrium's basic file browser that's accessed by clicking the blue computer icon in the Rescue Media taskbar. Does that show the drive? If not, can you open Command Prompt and enter "list disk" to see if Rescue even sees the disk? If it does, enter "select disk X", replacing X with the number of this problematic disk, then enter "list partition". Are the expected partitions of that disk listed? If so, enter "list volume". Is the partition on your external drive listed there, hopefully with a drive letter?
If you can't see the disk at all using any of those mechanisms, open Rescue Media Builder and go to Advanced > Devices & Drivers. Do you have the "USB override" checkbox checked? If so, try unchecking it in case this is a USB driver issue in your Rescue environment. That option was introduced to FIX some USB 3.0 driver issues that some people had, especially when using WinPE/RE 3.1 (Win7 kernel), which has no native USB 3.0 support, but after it was introduced, other users found that enabling it created some problems in their use cases.
In terms of the blue screen, it's probably caused by the CBT driver. As a workaround, if you can live without CBT for a bit, you can uninstall just the CBT component of Reflect until Macrium releases a patch containing an actual fix. To do that, go to Programs and Features, select Reflect, click Change, then select Modify, uncheck CBT, and continue through the wizard. Reflect works just fine without CBT.