I am not sure it is the SIM card as such. More like, a device with a SIM card inserted tries to register with the network. Using the cellular radio is what draws a lot of power.

Fun story I may have mentioned before. I had a Palm phone with a battery that lasted 9-10 days between charges. I did not bother to pack a charger when I was going away for just a week. My main usage pattern was texting and using apps (like games) that did not use the radio. Hardly ever any phone calls.

When the battery started getting old, I did not notice any change at all. It would still last the same 9-10 between charges, with the same usage pattern. The only change I noticed was that the voice in phone calls became distorted, due to the aforementioned power drain by the cell radio. Evenually it got to the stage when my phone rebooted when it received a call with the battery indicator below 90%. Replacing the battery fixed the problem.

The moral of the story is, do not trust the battery indicator. It may look like your battery is OK. The device may even behave like the battery is OK (see my 9-10 days between charges). But the battery may be already knackered. The OP's definitely looks that way.