There are tons of devices that do what you want but most aren't cheap because if you want cheap, just buy a receiver. Arcam has (or had) a 2 channel receiver with HDMI inputs. The Lyngdorf TDAI-3400 and NAD M33 both have optional HDMI boards. The Lyngdorf TDAI-1120, NAD M10, and Arcam SA30 all have single ARC HDMI inputs if you want to run your sources to the TV and send ARC back to the integrated amp.
All of the above also have room correction built in, which might help your setup since this isn't presumably going to be a dedicated listening space with perfect room treatments. You can also look at the Naim Uniti Atom or the new Cambridge EVO series, both of which have single HDMI ARC inputs, but no room correction.
I have the TDAI-3400 in our living room and the TV plays through it. It sounds incredible, quite honestly, for a 2.1 setup. I also have a 5.1 media room and while yes it is better for surround sound, it's surprisingly closer than you'd think. Actually, dialogue intelligibility, which is critical, is far better on the Lyngdorf than in my media room with a dedicated center channel.