How's the added latency when connecting a controller to the steam deck through Bluetooth?

I tried to do something similar to you without a cable (controller --bluetooth--> deck --wifi & steam play--> TV) but it had ghastly latency, yet I didn't isolate which leg of the trip was responsible.

I use mouse/keyboard primarily and never noticed an issue even with bluetooth. I don't play multiplayer so wifi is not a factor in latency.

I do have a USB wireless dongle for my xbox controller which apparently is faster than bluetooth. I also now use wireless dongles for my mouse/keyboard but mostly just for ease of use. All 3 USBs are connected via dock.