To start with, maybe just check the Offers and Wanted forum here and also place a WTB post: WTB, cheap Android box or media player with HDMI out for .ts and .MP4 files. Since you only want it as a simple media player, you don't need to worry much about specs, though you can do a lot more than just play media files on an Android box. I assume your TV has HDMI in. Someone may even give you something they aren't using for free. That happens a lot here.
All Android boxes will play MP4 files as far as I know. At least some, if not all, can also handle transport streams (.ts) so you might not have to convert anything. I have an old Android box running 7.0 and it plays .ts files without any problem. This would work fine for what you want.
The main advantage of a proper media player is that it will play a LOT of different file formats. I suspect that may be overkill for your purposes but I will look around for you.