I have this same headset and was in the same situation as you were, so I ordered this USB [https://www.flashbackltd.com/products/ps4-wireless-bluetooth-v4-0-usb-dongle|replacement adapter here] and it fully worked for me. It’s up to you if you buy it or not but if you do buy it and it does not work the website has a 30 day return policy on all electronics sold there. There should be a user manual inside on how to use it with your PS4, but because I threw away the box and manual so therefore I cannot provide instructions here.
The replacement dongle you mentioned is out of stock everywhere I looked, so that’s why I am reccomending a different one. So to answer your question yes it should work with your PS4.