Projector technology is crazy nowadays so you shouldnt have an issue if buying a cheaper brand new one for around $100. Just make sure the quality is at least 1080p or higher and you'll be good. Not sure if Amazon is available in the UK, but I just sorted by "Highest Customer Reviews" and picked out this one, was on sale for $89 at the time though. Amazon Projector I absolutely love it & always having it running movies, twitch or youtube on the wall whenever I'm trimming

Just make sure the projector and frame and parallel and square to each other. Distance doesn't matter so much, the closer it is to your frame the smaller your image will be and farther away the bigger it will be, so sometimes Im moving the projector closer and further to get the perfect size, and then once you find your size there are easy to use focus knobs to get your image crisp