bozidarms,
If I am understanding correctly, you attached an SSD to a USB adapter and then booted the DSO into its OS and copied the contents of the HDD onto the SSD. You were then hoping to be able to boot the DSO from the USB (change BIOS boot order permanently or just once), but when it did boot, you found that the drive letters were "wrong" - the OS was likely still seeing the original HDD and hence using that.
I would suggest, if it is possible to boot the DSO from a USB stick, that you grab yourself a copy of Ventoy to make a bootable USB stick that can boot multiple images and also a recent Clonezilla image... you'll want to be able to boot the DSO into Clonezilla. You'll want to make sure that the original HDD, the SSD (via its USB adapter) and the USBBoot stick are all present, then boot into Clonezilla. Use Clonezilla to copy the original HDD onto the SSD. The fact the normal DSO OS is not running and Clonezilla is running from a RAM disk means it should be possible to make a perfect copy.
If you only need a clone of the DSO's disk / OS then you are done at this point. If you want it to boot from the SSD then read on :
Once you have copied the HDD onto the SSD (outside of the DSO OS), you can then try to boot from it, but beware that the OS may spit its dummy out with a USB HDD. You need to "hide" the HDD from the OS - it should be possible in the BIOS to deconfigure the HDD (that will prevent it from BIOS booting, but the OS may still see it). If the OS still sees it then you'll need to disable the drive controller in the BIOS.
At this point, if you are lucky, then you'll have a system booting and running off the USB SSD, with its drive letters where they should be, without needing to open up the DSO.
It should be noted that this is ugly as it gets, and a FAR better option is to open the DSO and swap out the HDD for the SSD - you don't need to mess about with BIOS settings and you don't need an external device to boot the DSO. There is one gotcha here, depending on the BIOS it may barf at the new disk (if it is too big, for example).
If that happens then it's time to dig out your Ventoy USB boot stick again. You'll need to download a bootable HDAT2 image and pop that on the stick too, because you're going to need to tell the SSD to lie. Note that this almost certainly will not work using the USB adapter - it will need to be directly connected to a "real" disk controller. You'll need to query the total number of sectors / disk geometry from the original HDD, so boot one time with Ventoy with the HDD connected so you can get that info using HDAT2. Once you have that info it's time to reboot to Ventoy / HDAT2 again, this time with the SSD directly attached so you can update the SSD.
If the BIOS won't even POST with the SSD plugged in then don't worry - with the SSD unplugged, allow the system to boot to Ventoy to get past the POST, then (hot) plug in the SSD. Fire up HDAT2 again and use that to change the reported disk size. It doesn't matter that the BIOS hasn't seen the SSD, HDAT2 will do a fresh scan.
Finally, with the disk reporting the same geometry as the original HDD, it's time to reboot, one last time.... into the DSO's own OS, nice and quickly with no spinning rust, from the now-internal SSD
Hope this helps,
Pat.