Yes, if you want a Bluetooth headphone it should support aptx LL at best. Here is a list.

https://www.aptx.com/product-listing?product_category=7&aptx_type=3&created[min]=2019/09/31&created[max]=2022/12/31&sort_by=created&sort_order=DESC

Don't use a Bluetooth transceiver as a headphone amp. Less power and features, too big, volume control sucks.... . Better stick to your original idea and use the 1mii transmitter and a btr5 or qudelix 5k. (maybe btr3k if on a tight budget)

The EQ function to adjust the sound to your liking is also a big plus.

One thing, you have a 5.1 system connected to your TV? Then you would have to change the tv settings to PCM for headphones and back for 5.1 all the time.