The particular file I need (and can read in Quick Look) is a Numbers file.
Ok. that should be easy to recover.
The original backup was created under an earlier version of the O/S.
Does that mean you have continued backups after update to Mojave using the same Time Machine backup?
If you have not used it in mojave you must inherit it first.
Time Capsule shows as 'disconnected'.
That appears to be more to do with the sparsebundle not being readable. And I would suspect permissions issues.. since it is not allowing you to save the file or recover it.
Permissions got some big upgrades in later Mac OS and it does cause issues with accessing older files in a backup.
Apple removed the main permissions tools from disk utility which makes life that much harder.. there is a trick.
This is a bit complicated.
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 launcher.. you need older Mac OS.. early Lion.. or special script from internet I cannot post URL but just google the text in bold.. still works even for mojave.. although it was not so straight forward. Any windows computer you can use.. or bootcamp partition on a Mac you can use the windows version of the airport utility.
If you 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. Very fast.. takes just a few seconds to run through the disk and open up access. It might offer one easy way out of your problem.. !!

If this is the only source of your file.. then of course I would recommend you backup the TC before you do it.. there is archive function in the airport utility so you can save the current backup. But this has worked whenever I have needed it.