This is a problem when I had to switch phones. Last Pass Authenticator will save the TOTP secrets in the cloud. Let's say you lose your phone, the following will happen when you try to install Last Pass on the new phone (note: I am talking about Android, but IOS is probably similar).
1. You tried to install Last Pass Authenticator first because you know that any new device you add to last pass will require 2FA, but when you tried to do this, the last pass authenticator won't work unless you install Last Pass first.
2. When you install Last Pass on the new phone, it request that you authenticate using 2FA. Unfortunately, the authenticator on your new phone won't be active until you log into last pass, so you would need to authenticate using the old phone that you no longer have.
The safest solution would be to enable the Grid in Multifactor authentication. When last pass request 2FA, you select the alternate method of the grid and use the paper copy to enter the 2FA code. Once you are in, you can go and restore the secret to the last pass authenticator.