Ok.. You don't have to wipe since you have so much free space. And I understand you want to keep a backup with all of the files intact that you might have deleted on the way to saving some space.
Connected directly should help.. depending on which model TC we are talking about .. latest one and close by.. for sure.. you should get high speed over 5ghz wifi.
Let me recommend a couple of things.
Firstly I would archive the existing backup.
Plug a Mac (HFS+) formatted USB drive into the TC and run archive on the disk tab airport utility. This is important because Catalina has shown a nasty predisposition to wiping out or corrupting earlier backups. You should only need a 1TB disk although normally you would use same size as the TC.. ie 2TB or larger. Since your data use on the disk is less than 1TB it should fit.
Second step depends.. I think I would do a factory reset.
NOTE.. factory reset does not delete any files on the hard disk.
After the reset configure the TC again using the airport utility in the Mac.
DO NOT USE APPLE RECOMMENDED NAMES.
Use short names, no spaces and pure alphanumerics. Bonus for less than 8 characters. Use same rules for passwords but longer.. 8-20 characters.
Test from Finder you can now access the TC.
You will need to point Time Machine to the newly renamed TC location. TM will indicate it is going to start a new backup but will discover the backup already exists and after checking complete the next incremental. If it refuses to finish again.. erase the TC disk and start again. You will have all the old files in the archive disk.
Third step .. I would create a clone. The best place for this is USB drive plugged into the computer.. same size as the disk in the computer or larger. I use Carbon Copy Cloner.. it if free to use for a month so give it a run. You might be very glad of it. Set it to bootable.. and test boot from it.
You can also use CCC to create a clone on the TC but it is not bootable.
Now you are ready to do Catalina.. and if you have trouble just boot from your clone.