I recognise the BSR deck because it's the same as one I used to own when I was 12. The cartridge body is built in to the headshell. The needle is that bit that says ST16. The LP.S bit is for playing regular singles (45's) and LPs. The needle is double sided. If you flip it through 180 degrees so that the legend on the other side is uppermost when fitted in to the arm, then from memory (35+ years ago) I think it might read 78. If it does then for heaven's sake don't use that on your vinyl. It'll plough a new furrow!!

It's safe when you can see the LP.S legend when looking down on the deck.
The 'needle' (stylus) is
cheap as chips; around £6 to £9. The styli you have might be okay, but just in need of a clean. However, the chemicals and microscope required for the work will cost way more than £9, so for the sake of it if you're going to all this trouble then I'd chuck a tenner at a new stylus just to be sure you're not dragging an old dirty diamond around your record collection.
The trickiest thing in this plan to divert the wiring is going to be working out the grounds and signals. It's easy to get the polarity of one channel inverted, and that will mess up the sound.
If BSR followed the convention, then the wires should be colour coded. The signal colours are Red for right and White for left. These go to the centre pins on each phono plug. The grounds are a little trickier. Red ground is Green. Left ground is Blue. These go to each ring connection on the phono plugs. Cables can lose a bit of colour definition with age, so a quick wipe over with a damp cloth should help. If you are red green colour blind then get some help with this.
If you look at the pivot of the arm there's a screw adjuster. This is for the tracking force. Unless you have scales that weigh in 1/100ths of grammes then leave this alone. The box with the stylus probably gives the tracking force, but without something to measure it then just leave things where they are.