Alphard key battery low warning

Indeed, a 'key battery low' warning combined with the symptoms you describe strongly suggests a depleted battery in your electronic key. The key fob uses a small battery, usually a CR2032 type, to transmit the signal required for keyless entry and starting. When this battery weakens, the signal strength diminishes, hence the intermittent or complete loss of functionality. It is very likely that this 'low voltage' is what you are seeing.

Thanks! So, just to be clear, you reckon it's almost definitely the key fob battery and not something more complicated with the car's receiver? Is there any way to be absolutely certain before I start tearing into the key fob?

While other issues could theoretically cause similar symptoms, the 'key battery low' warning makes a failing fob battery the most probable cause by far. The fact that you already have the low battery warning strongly indicates that replacing the battery will likely solve your problems. However, if replacing the battery does not resolve the issue, a visit to a qualified workshop is recommended to assess the vehicle's receiver and associated systems.

Just wanted to let you know, I replaced the battery in the key fob, and that seems to have fixed the problem. Total cost was 37€ for the new battery. Thanks for the help!