I’d just buy one of these for 21 quid and add in a cheap 2nd HDMI cable if you need one. Looks like it’ll give you the audio to plug in something else and the video for your projector?
(Note that is just the first one I came across - it’s a pain to google for the UK because of my Aus location, so there may be cheaper but I like how sturdy that one looks, tbh.)
Edit: the thumbnail is for the slightly cheaper one but that only outputs (it looks like) audio as optical. The £20.99 one definitely seems to have a standard audio jack.