But it had file copied to it that were sitting there "beside" the sparse bundle" file.
The picture is getting messy..
From the above comment it seems you directly copied files from the Macbook to the TC hard disk.
Did you then upgrade the laptop to Catalina.. as that is bound to cause issues??
using Migration Assistant with Catalina might restore the permissions and restore access to my files. Does that sound closer to a solution?
Migration Assistant will copy the files from Time Machine backup to the computer. It will not copy anything outside of the Time Machine backup. Did you delete the files copied to the TC from the Macbook? If so they are not included in the last backup even if they are in the history. Migration Assistant cannot restore files from history .. only from the last backup. Time Machine cannot backup files located on the target drive or any network drive.
Recovery of these files outside of the Time Machine Sparsebundle will require manually copying them after changing permissions.
Let me suggest a few things.
#1 It is really hard to get permissions to change on a network drive. And if the files have no backup you should do an archive of the TC now. Archive will use a USB drive of equal or greater size than the TC disk and HFS+ formatted. It will take up to a day for 2TB. More for 3TB.
#2 Plug that USB hard disk into any computer and you should be able to fix permissions directly from Finder, and copy the files to the computer.
#3 You can use migration assistant to recover files to a computer running catalina.. although it may take a bit of work to do it and migration has its own issues.
Read the Apple documentation on migration.
https://support.apple.com/en-us/HT203981
And just leave the Mac Pro out of it since it was never an originator of backups to that Time Capsule anyway?
That is up to you.. migration requires the computer is running the same or later OS than what you used for the backup.
#4 There is another way if you wish to use the Mac Pro and that is Time Machine in Recovery and recreate the boot disk of the laptop as it was before you lost the files you want. This can be done from any computer so whichever is most convenient and has the resources you need.
#5 The is a trick to get around permissions I explained in other threads.. but you need to use Airport Utility 5.6.1
YMMV and a backup of the contents of the drive is wise if there is no other backup to these files.
There is a permissions fix in the old airport utility where you can add guest access.
Your first problem is to run 5.6.1 airport utility.. you need older Mac OS.. early Lion.. or special script from internet I cannot post URL but just google for
Launcher for AirPort Utility v5.6.1
It still works for me even on High Sierra.. although it was not so straight forward. Any windows computer you can use.. or bootcamp partition on a Mac. Mojave and especially Catalina are unlikely to work even though the download is available.
Once you access 5.6.1 airport utility on any Mac or Windows computer.. does not matter which.. go to the Disks tab, File Sharing.. by default Airport Disk Guest Access is Not Allowed.. change it to Read and Write.. and click update.
The change will take just a few seconds. This may have multiple affects I cannot predict. But it does offer one easy way out of your problem.. IF IT WORKS!! Many people have tested it and yes it does work.. but take care this is not your only source of the files.
To gain access to files copied directly to the TC this is my top rated method.. however dirty it looks.

NOTE.. once you recover the files.. you have now changed access permissions and really should erase the drive and start over.