The two batteries I'm testing out can both handle around 30mA continuous output and are working perfectly.
Got a 40mAh LIR2032 Li-Ion coin cell and a 15mAh 3/V15H NiMH coin cell (thought I'd bought an 80mAh - oops).
The LIR2032 has been running since Friday morning and the 3/V15H since yesterday afternoon and the voltage hasn't dropped yet. Looking good.