I'm sorry to hear about the issue you're experiencing. I'm not sure what's happening, because every part of Face ID is handled by iOS and not mSecure. When you open any app, the app simply tells iOS, I need you to authenticate this user, then everything from that point on is handled outside of iOS. After the user is authenticated, then iOS hands back the password so the data in the app can be decrypted. So as far as the actual recognition of your face goes, there's nothing mSecure can do to make that part better or worse.
For the other issue, its sounds like mSecure's permission to use Face ID has been removed from iOS. To fix that, open the iOS Settings app, scroll down and tap mSecure, then make sure the "Face ID" toggle is turned on to allow mSecure access to the feature.
This is a very strange issue, because there should be no timeout for the Face ID feature where it stops getting displayed in mSecure's Settings. I just tried having my wife unlock mSecure with her face, which doesn't work on my iPhone, and even though mSecure tells me the her face isn't recognized, it doesn't remove Face ID from the Settings after I open mSecure with my password. Are you saying that you see a message from Face ID saying "Face ID attempts exceeded"? If so, how many times do you attempt to unlock the app via Face ID before you see that message?
Also, when the Face ID toggle isn't available in mSecure's Settings, what do you see when you open the Settings app and then scroll down and tap "mSecure"? Does it show you that Face ID for mSecure is disabled?