Disabling keychain doesn't let me use Face ID with the Southwest Airlines app

I may be wrong, but I think your right on both.
Go to iPhone settings then scroll down to the SouthWest app and tap it.
At this point do you see face i.d. enabled for that app?
If not then enable it.
Next sign into Southwest app and look for settings inside the app itself, because not only should face i.d. be enabled on the iPhone it should also be enabled on the app. In the app settings look for security or login password or biometrics until you find face i.d. or touch i.d. and make sure to enable there as well if you can find it.
I have seen some apps still say touch i.d. even though it will work with face i.d. so look for that too.
You might not find it unless you sign completely out of the app and remove the username as well and see if you see an option for touch or face i.d. before your tell the app to remember your username. Their may be an option to put a check in touch or face i.d. before you enter the username and app password.

You want the iCloud keychain ( not to be confused with Safari keychain) to be enabled for sure in all cases weather you use auto fill or not!
iPhone settings, name at top. iCloud, passwords and keychain. Make sure it is enabled.
The reason you want the iCloud keychain enabled is, because it uses end to end encryption like 1Password and you never know when it might be used if you restore a backup to a new iPhone the iCloud keychain will help restore app passwords in some cases, but not all in apps.
1Password even uses iCloud Keychain to store your 1Password secret key which is safe to do so.
Still save your secret key in a hidden place just in case.

As a last resort if the above does not work.
Personally I just have 1Password selected for auto fill, but if you can not find to enable face i.d. inside the Southwest app then I would try by enabling both 1Password and keychain for safari and try that.
It want hurt to enable both cause it will just give you an option. By enabling keychain for Safari your gonna save your Southwest login there.
Personally I use 1Password and not Safari for saving my passwords.
Maybe the guy at Southwest meant you have to use keychain for Safari auto fill to use face i.d. through the browser, because maybe it is not available through the app.
I have never had an app require to use face i.d. that way so not sure.