This is the code I ended up using (for now):
Sub CopyBordersOnlyLV()
Dim myRangeC As Range
Dim myRangeP As Range
Dim i As Integer

ActiveWorkbook.Styles("Normal").IncludeBorder = False

'The following lines store a selected range of cells in the variable "myRangeC"
'which will be the range used to copy the border linestyle from:
 Set myRangeC = Application.InputBox(Prompt:="Select Range", _
        Title:="Select Range with Borders to Copy", Type:=8)

'The following lines store a selected range of cells in the variable "myRangeP"
'which will be the range used to copy the border linestyle to:

PasteRange:
 Set myRangeP = Application.InputBox(Prompt:="Select Range", _
        Title:="Select Range to Apply Borders", Type:=8)

 Set myRangeP = myRangeP.Resize(myRangeC.Rows.Count, myRangeC.Columns.Count)

For i = 1 To myRangeC.Count
    myRangeP(i).Borders(xlEdgeLeft).LineStyle = myRangeC(i).Borders(xlEdgeLeft).LineStyle
    myRangeP(i).Borders(xlEdgeTop).LineStyle = myRangeC(i).Borders(xlEdgeTop).LineStyle
    myRangeP(i).Borders(xlEdgeBottom).LineStyle = myRangeC(i).Borders(xlEdgeBottom).LineStyle
    myRangeP(i).Borders(xlEdgeRight).LineStyle = myRangeC(i).Borders(xlEdgeRight).LineStyle
 
    myRangeP(i).Borders(xlEdgeLeft).Weight = myRangeC(i).Borders(xlEdgeLeft).Weight
    myRangeP(i).Borders(xlEdgeTop).Weight = myRangeC(i).Borders(xlEdgeTop).Weight
    myRangeP(i).Borders(xlEdgeBottom).Weight = myRangeC(i).Borders(xlEdgeBottom).Weight
    myRangeP(i).Borders(xlEdgeRight).Weight = myRangeC(i).Borders(xlEdgeRight).Weight
Next i

Dim Answer As String
   
 Answer = MsgBox("Do you want to continue?", _
            vbYesNo + 256 + vbQuestion, "Continue Pasting Borders")
    
 If Answer = vbNo Then Exit Sub
    
 If Answer = vbYes Then
    GoTo PasteRange
 End If
 
End Sub