105 immobiliser issue

If you bypass it you wont be able to shut the engine off.

Google search:

landcruiser 105 bypass fuel solenoid

They tend to fail once warm. In the short term, bucket of cold water on the pump can help get it started.

The immobiliser is a prick to remove, I used a flexi-head Dremel to grind out the tamper-proof bolts then cut away the unit. From there its just a matter of joining the two wires somewhow.

It cannot be bypassed while attached to the fuel pump.

Jacob
In the interim, if it is getting hot and stopping the engine or not connecting while hot, try using a length of vacuum cleaner hose from front of engine bay and AIM the forced airflow onto the pump area where it is located. That way the unit will be cooled to some degree and may begin to function normally again. I have used similar on Hall Effect ignition sensor units in petrol vehicles to keep them cool. Never had a failure after that hose was placed there!
Some have used a small computer fan to provide a cooling flow!