If you use Migration Assistant to transfer the data, it will create a new user with your data, so then you will have to transfer the data manually to the user you want, or keep this user and delete the old one.
In this case, it's much better to transfer the data manually if you want to show it in one user that already exists. Simply put the files in the folders where they are in your old MacBook Pro
There is no need to remove your existing account unless you want it replaced by the migrated account. You cannot manually transfer files from a Time Machine backup. You must use Time Machine to restore even individual files. So migrate the old account so that you have two separate accounts.
If you wish to then transfer files from one account into the other you can do that. See Transferring files from one User Account to another. When you have your one account as you want it, then you can delete the other account.
I have a similar issue transfering files from a MacBookPro to new MacMini. Migration Assistant transferred some files across from Time Machine without any problem, however documents, downloads, iPhoto and iTunes Libraries all have 'no entry' signs against them. Migration Assistant has created a new account so I am wondering how I can transfer and access these files and programs?