I use wired headphones to study with Anki (AnkiDroid) because I've found most (inexpensive) Bluetooth headphones require a second or two to begin playing. As I'm dealing with short audio clips, this use case necessitates restarting the "audio playing" situation every few seconds.
Maybe the app developers could "play" quiet audio between these short clips. But barring such a development, I'd like to know if higher quality headphones might suffer from less latency in this regard.
1-2 seconds is an eon for audio latency so I guess something else is going on than anything BT related in the headphones. Unless you have particularly bad luck in what headphones you use.
FWIW, I use a variety of cheap and not so cheap BT headphones across multiple devices and apps including AnkiDroid and have not perceived any latency.
If switching to wired removes the latency then it does seem to indicate something in the BT stack of your device. I wonder if you experience the lag when using AnkiDroid + BT on another device.