It's a bit unclear what you're about to migrate from where. And why you'd lose access to the existing project. And you have different options based on the current source control type selected.
One option which you could try is to create 2 new accounts and leave the whole old account in read-only state. That should leave the history available to everyone. You can then spin up as many new accounts as you want, using just the latest version of the sources.
Git
If it's a Git repository it's as simple as making a local clone of the whole repo, creating a new team project in VSTS and pushing the clone into its second home.
TFVC
If it's TFVC, it's much harder. I've used OpsHub in the past which works reasonably well, but in our case completely got stuck in a couple of strange merge situations. Those were probably created as part of work done back when that team project was hosted in TFS 2008, so you may be luckier than we were.
You could decide to move to Git as part of your migration. Use git-tfs to create a local git repository with all your TFVC history and then push that into a bare Git repository in your new team project. Or use the TFVC import tool. There's quite a bit of documentation on this subject.
The VSTS Sync Migrator supports a snapshot without history as far as I can tell. Which would not suit you.
VSTS Copy Project doesn't support TFVC, and is no option in this case.
An option that's missing from your list is Timely Migration, it supports TFVC to TFVC migrations among other options. I've used them a long time ago to copy data between TFS servers. Back then they were working exactly as advertised.