Diode drop mode seems to do it. With auto on both sampling settings it was extremely slow (second channel at internal temp, so 125Hz/64smpl = beep delay seems like seconds).

If i set sampling to 8kHz/32smpl it gets about acceptable. There is still bit of a delay, but it can (maybe, not measured) detect something like 100 milliseconds of short circuit.

Some quess work:
32/8000 = 4 ms set of samples, but processing+bluetooth takes just under 100 ms between sampling. ~2-100 ms minimum contact time might be enough for beep, unless it actually needs two trigger events back to back, so then it might be closer to 100-200 ms.

If I just bounce test leads to each other, it seems to get some pretty quick touches and beep. Then there is that delay, sound would not start at the moment the probes touch, BUT it does beep after you have disconnected, so it is hard to miss that you had a connection just hundreds of milliseconds ago.

I just tried to give it a go with real time application and checked which contacts on a relay board are NO or NC, i got reliable beeps and could not move probes any faster. In this case mooshimeter did the job way better than i first thought (as i had probably tried it in the past with just automatic sampling settings).

Wish for better beeps, should be possible on the app side:
It would be easier to find, and more reliable to use, if beep-mode had its own thing on the mode selection, where it would set max sample rate with least samples per round to give it some real time feel. Diode drop is so different application that most of us wont know it has a beep mode in it and those auto-settings for it wont do what we need for beeps.