High efficiency pwm current limiter for charging supercap from alkaline battery

I’m designing device which uses two alkaline C-batteries in series as power source, and sends SMS messages using SIM800. On average 1 message per day is sent. Goal is to maximize battery life.

Allowed operation voltage for SIM800 is 3.6V-4.4V, and peak current around 1000mA while sending. Alkaline batteries are known for NOT having good recovery effect with pulsed discharge, so plan is to limit the current around 50mA-100mA and slowly charge the 1F supercap, then using power from cap to send the message.

The whole idea of using current limited charging is to improve overall battery efficiency, so obviously current limiter should be as high eff as possible.

I figured to put the current limiter to low-side, as N-channel mosfets are better in general, lower Rds, lower gate charge, so better efficiency and can be driven with MCU gpio as is.

I came up with a low-side design like this:

There is MCU on board, and PIC[1..12] bus goes there. VCAP_MEA measures low side voltage and VCAP_DRV drives the mosfet. There is no current sense, but as ESR of C1 and coil L2 properties are known and pretty much static, i figured i can make it close enough with measuring the voltage and ramping up the PWM as voltage rises..?

For coil L2 i planned to use NR6028T101M.

Vdd is 4.3V and comes from TPS61023 based dc/dc boost converter, 1.6V-3.6V from two alkaline cells is boosted to a 4.3V.

The problem is, i have very little to no experience with custom switching circuit designs. Would that design even work as planned?

What would the efficiency be, does it make any sense to use approach like that or am i better off getting the 1A peaks straight from the batteries?

How will switching frequency affect the efficiency? I did some simulations with 100kHz and picked the coil accordingly, so that the simulated voltage looks reasonably smooth.

Part descriptions:

Coil L2 – NR6028T101M – 100µH Shielded Wirewound Inductor 660mA 780mOhm Max
https://www.digikey.fi/product-detail/en/taiyo-yuden/NR6028T101M/587-2105-1-ND/1866245

Mosfet SSM3K127TU,LF
https://www.digikey.com/product-detail/en/toshiba-semiconductor-and-storage/SSM3K127TU-LF/SSM3K127TULFCT-ND/10380188

Capacitor DGH105Q5R5 – 5.5V, 1F, ESR 260mOhm.
https://www.digikey.com/product-detail/en/illinois-capacitor/DGH105Q5R5/1572-1771-ND/7387509