I've been using the japanese style Bluetooth Saturn Pad from RetroBits for about one week now and I'm blown away by how good this controller is.
I'm mainly playing on switch and can't notice more input delay than using a pro controller.
Playing Thunder Force IV with this controller on my switch feels like a dream.
Absolutely no complains about the awesome dpad and the shoulder buttons so far.
I can also confirm that this controller can be used on a PS4 when using a white Brook PS3 to PS4 Super Converter as it has PS3 Support built in.
The battery also seems to last very long, as I've been playing for almonst 15 hours by now without recharging.
Also super nice that they have implented rumble but it is obviously not as good as true HD Rumble of an original Switch Controller... maybe the rumble is a bit to strong sometimes as well but many games can adjust the strength via the options.
They also implemented a nice little button combination to swap A-B and Y-X Buttons that makes it super easy to play some games with the default button configuration without having to adjust the button config manually.
The only issue I had at the beginning was that it wouldn't pair with the switch easily you have to be like really aggressive during the paring by mashing the home and the A Button like a mad man. It seems like the pairing is too short or something so that you need to extend it by mashing... just dont give up mashing

I hope they'll fix that in a future firmware update.
Initially I had a hard time deciding between the Bluetooth and the 2,4 ghz versions but for me I am addicted to my switch and want to play as easily as possible without having to connect a USB dongle.
As far as retro systems go Retrobits will be releasing receivers for the bluetooth controller as well so that we can then play on original systems (Genesis and Saturn) with the Bluetooth Models in the future as well just like we can now with the 2,4 ghz models.
Hope this answers some of your question marks you might have in regards to these awesome controllers from retro bits.
Cheers
mo