get 100mA pulses from 20mA source to charge li-ion
I think pulsing charge to a Li-ion battery isn't a good idea, since they have a limited number of charge cycles. I also doubt that pulsing a high current gives faster charging than constantly charging at a current your power supply can provide.

The datasheet for TP4056 says "absolute maximum rating -0.3 - 8 V", so I wouldn't recommend to power it with a nominal 8 V that might fluctuate higher, if that's your case. The datasheet also specifies a programmable current between 130-1000 mA. Since you only have 20 mA @ 8V, I'd look at some other charger that can provide lower charge current, for example the common MCP73831.

I would also look at some regulator to get 5V from your 8V. The choice depends on the requirements of your project. Some reasonable choices are a linear regulator, a ready-made switching module (such as from Pololu) or a switching regulator IC with your own supporting components. A switching regulator would be most efficient in your case.

This suggestion to regulate the voltage plus programming a charger to stay within your power budget will give a simpler and more reliable solution than your original idea.

Alternatively, LTC4002 seems to be more efficient, as you say. At a first reading, I don't see what charge current limit that can be programmed.
EDIT: The current is limited by Rsense, with the formula I = 100mV/Rsense. I don't see what limits there are and I'm not clear what will happen if LTC4002 tries draw more current than available.