Maybe not the answer you are looking for, but what TV are you looking at?
Most smart TVs support streaming film services, and any films you want to stream from the laptop will play over DLNA, using something like servio..
My media center computer rarely gets turned on anymore, my TV will play everything from the server including HD content and it's not a current Gen smart TV, and I have a server stored away out of sight (noise isn't an issue then) and any computer would do for that, I'm running an N36L HP micro server for this purpose without issue...
It's much easier than having a separate computer, mouse/keyboard etc..
..Just a thought and then you could spend your cash on a nicer TV then
Edit: other option to look on eBay or here for some second hand kit to reduce your budget, or look at the option of a HP microserver £110 after cashback, but you'll need to get a graphics card on top of that.