You need ceter tap to do a full wave rectification with two diodes vs.
4 without center tap which is bridge rectification.You did not have to use exact OEM one, any x-former with same spec. would do. Only matter is physical dimension for fitment on the board.
Again no meed to get OEM one. Take the battery to battery shop, they will give replacement with same rating and size. If terminal is different size, you can just solder wires onto it. Or replace clips on the wires.