If you insert a new sheet and paste the contents of the "basic" sheet as it is, you will get the following contents.

  Worksheets.Add After:=Worksheets(Worksheets.Count)
    Sheets("Basics").Cells.Copy
    Sheets(Worksheets.Count).PasteSpecial
    Application.CutCopyMode = False

Excel
The width of a column depends on the column and the width of a row depends on the row (it is not possible for individual cells in the same column to have different cell widths), so if you want to keep the column width the whole column If you want to keep, you need to copy the entire line.
If both, copy the sheet itself, or
I think there is no choice but to set rows and columns individually.

Addendum

You can copy the entire cell.
It is a scale from the eyes.

If you want to operate manually, select it and tell Excel,
In VBA, you will be instructed in writing what you want to operate, so
It is not necessary to write instructions for selecting and activating.
It's a useless operation.

Sub test1()
    Dim wshTemplate As Worksheet'' Template Sheet
    Dim wshNew As Worksheet'New sheet
    Dim strName As String'Sheet name
    '' Preparation (arrangement of prerequisites)
    strName = Worksheets("Customer Management").Range("D1").Value
    With Worksheets
        Set wshTemplate = .Item("Basics")
        Set wshNew = .Add(after:=.Item(.Count))
    End With
    '' Post&add
    With wshNew
        wshTemplate.Cells.Copy .Cells
        .Name = strName
        With .UsedRange
            .Cells(.Rows.Count + 1, 1).Value = strName
        End With
    End With
End Sub

However,
1) Create a new sheet
2) Copy and paste the cell of the template sheet
If so, the work will be two steps,

1) Copy the template sheet

Does one process produce the same result?
Changing the sheet name is the same.

Sub test2()
    Dim s As String
    Dim ix As Long
    With Worksheets
        s = .Item("Customer Management").Range("D1").Value
        ix = .Count
        .Item("Basics").Copy after:=.Item(ix)'← copy the whole sheet
        With .Item(ix + 1)
            .Name = s
            .Cells(.Rows.Count, 1).End(xlUp).Offset(1).Value = s
        End With
    End With
End Sub

If you have any commentary, please ask.

I tried it with a macro recording

Sub Macro1()
'
    Cells.Select
    Range("A3564").Activate
    Application.CutCopyMode = False
    Selection.Copy
    Sheets("Sheet1").Select
    Cells.Select
    Selection.PasteSpecial Paste:=xlPasteAllUsingSourceTheme, Operation:=xlNone _
        , SkipBlanks:=False, Transpose:=False
End Sub