Alternative for the DF Player (mp3 playback module)?
You must be not the only one being concerned about this issue; good designers and manufacturers usually listen to their customers and improve their products accordingly.

I think device you refer to is based on YX5200 chip, which has rated current of 20 mA. There's newer device, YX5300, having rated current of 10 mA and standby current of less than 200 uA (as declared in the datasheet). I think this is the device you need.

However I am concerned about the responsibility of the manufacturer - datasheet does not state its name and contact details; it does not state source of the intellectual property used. Most probably someone is missing their royalties, and buying this device you are participating in counterfeit - e.g. they glued several ready-to-use SoC parts together, and used freely available C source code, which most probably has requirement in its license for the source code origin declaration in the documentation for the end product.