Many thanks for sharing your project. It works like a charm.
I chose the WEMOS D1 mini as base and stacked the wee OLED shield on top. I had to change GPIO 4&5 to 2&15, because this display is controlled by I2C bus.
Next I bought a small .96 colour OLED, controlled via SPI and a SSD1331 controller; so switched back the GPIO lines again. Unfortunately, the reset via Vcc is not sufficient for this controller - it needs a decent reset after power up, so I had to sacrifice GPIO 0 for this purpose and so am stuck with the 5 button control on A0, but it's ok as well.
In addition, I use a matching WEMOS battery shield, which has a loding circuit for LiPo as well as a booster to rise the LiPo voltage to 5V.
By now, the circuit is drawing 120mA without amplifier - I'm using an external boom box right now.
Thanks again
Klaus