Powering SIM800C with Lithium polymer batter

Hii There!
I am hobbyist and developing one project where I need to SIM800C GSM module to be active 24/7. So ofcourse I thought of connecting a battery, I have chosen a 3.7V Lithium Ion Battery to the module directly keeping 100uF Tantalum capacitor accross it.

I am using TP4056 Charging module to charge the battery and also the output of tp4056 is driving my PIC16F microcontroller and GSM module.
My PIC16F is running at 4MHz and having voltage range of 2.2V go 5.5V. GSM SIM800C needs the voltage between 3.6V to 4.2V Max.
(Sometimes work even with 5V)

Now my problem here is the TP4056 is unable to charge the battery and power the GSM module simultaneously. Can any one suggest me how exactly this to be done? Or is there any completely different method I can apply?

Plz let me know. Stuck with the last step of hardware design. All types of suggestions are welcome! Thanks.