How do I transfer shares between accounts with all of the tax lot data included in the transfer?

If you're moving securities from one brokerage to another the proper way to do that automatically in Quicken is to use the "Shares Transferred Between Accounts" action that's available to you.  That action will create one "Remove" action for each security in the old Account and multiple "Add" actions in the new Account, one Add for each existing tax lot.