1Mii Bluetooth 5.3 Transmitter Receiver for TV to Wireless Headphones, Dual Link aptX Adaptive/Low Latency/HD Audio, Aux Bluetooth Audio Receiver Adapter for Home Stereo, Airplane, Boat, Gym

You need aptX Low Latency transmitter. Something like this: https://www.amazon.com/MiiLink-Bluetooth-Transmitter-Receiver-Adapter/dp/B08G8Q7NXZ/ref=mp_s_a_1_3?dib=eyJ2IjoiMSJ9.UagICCTGd_hlZcVKILPiIs8hkENe2RQbwu9XwROD8dmS8snXBimtZuXlDb3F2ty8TEKGoDDtdhH2xGQ5o1UH0anc5I9G0rFw6hCaLkHjBFHX-gx61AIhgYAcekc6pYiMM7uTP1f8pG7PZj-I3_mzWv5_qeqpfyNlfkWSQK9JgXpxo0k_eXZg0vZ0P5M3bDwUH-OTRmy47WcFLyCv2wpVIg.gyhp3__UF-6VXzI6okO4TYILVWpdPgLOTRiop9Sb-Tk&dib_tag=se&keywords=aptx+low+latency+transmitter&qid=1782831273&sr=8-3

Before my iPod was interred on it's Golden Throne to keep working locally, I've used one like this on the 3.5mm port onto it to pair it with both bluetooth earbuds and speakers. Bluetooth transmitter/receptor

You can use a combo of an HDMI audio extractor and a Bluetooth transmitter to accomplish what you want to do. (If you aren't playing at 4K then you can use a cheaper 1080p HDMI audio extractor.)

I haven't bought one since like 2020, but I have two that look like this one, and cost about the same, so they're likely the same circuitry: https://a.co/d/0hQU9z5w They've been great (they have an internal battery so you can use it in the car or whatever). You might ask/search over at r/budgetaudiophile to get a better suggestion.