After some frustrating nights, I finally have the SPV1040 solar charging IC working with my Iowa Thin Film Technologies photovoltaic strip. The PV puts out a nominal 3V at 25mA. Basically, the chip requires two voltage dividers for voltage and current feedback. The ratio of these dividers determines the output.
While I have try and squeeze more out of it, the IC’s 100kHz PWM is able to knock this up to 5.5V at a peak of nearly 3mA. This is enough to power the TP4056, which is designed primarily for USB-powered charging. Given the relative gluttony of the charging IC, full sun is needed with the single PV strip in order to attain charging current. STM’s eval board uses a more efficient LiPo charging IC, which is something I am planning to investigate in my quest for efficiency.