SIM800L with Three SIM not registering to network

Using Arduino Networking, Protocols, and Devices
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Here in the UK, Three do a brilliant free SIM offer with 200MB/month, so I got one which I'm trying to use in a SIM800L module for an IoT project to help my mother-in-law who suffers from dementia. I just need it to receive texts and do minimal Internet access.

I wonder if anyone else has successfully used one of these free Three SIMs in a SIM800L?

The Three SIM works fine in a Three data dongle. I can access the Internet and receive texts. But if I put it in the SIM800L the LED blinks once a second indefinitely, indicating that it's not registering to the network. AT+CSQ typically gives a signal strength of around 17.

But if I put my Giffgaff SIM from my iPhone in the SIM800L the LED blinks once a second for around 7 seconds, then once every 3 seconds, indicating that it's registered to the network. (Giffgaff runs on the O2 network.) AT+CSQ typically gives signal strengths of around 19.

I'm supplying the SIM800L with 4V using a MIC29302 low dropout linear regulator, and I've tried feeding it from 2 different 2A capable 5V USB PSUs. If I place a 2 ohm resistor across the 4V output (to draw 2A) it drops by only 0.06V (until I drop the resistor, which is too hot to hold after only a second or so!)

I can exchange AT commands successfully through an Arduino (currently, a MicroView) running at 5V with Rx and Tx connected to the SIM800L through a bidirectional logic level shifter, the low voltage side of which is supplied at 2.8V.

Is there some special parameter you have to set up with the Three SIM in order for it to connect? Is anyone successfully using a SIM800L on any Three tariff?

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In fact I've now established that the Three UK network doesn't support 2G. It did so previously only through agreements with other suppliers but evidently they decided it was a market segment they could afford to walk away from.

I now have to make a choice: (a) a much more expensive 3G module, (b) a bargain basement PayG SIM, (c ) use a USB mobile dongle with the free Three SIM and a Pi rather than Arduino. But my original objective of a very simple and cheap SMS to TV IR code gateway for a dementia sufferer with no ongoing fees seems out of reach.