After weeks of deliberation, I bought some batteries from eBay, followed the guide (also watched a few youtube videos of the same project). And finally did this project. The instructions and comments really helped, so make sure you read them all.
I didn’t solder, just used thin strips of electrical tape. I also reinforced the white wires that join to the board, so that they didn’t break while manoeuvring everything around.
The end result was a perfectly working shaver, best AU$15 spent. Managed to save a device going to landfill and save AU$120 in the process.