How about a shunt resistor for small power loss + opamp?
A 10mOhm shunt will only drop 100mV at 10A, seems way more than your application requirement, so actual drop will be even lower.
A dual 8-pin package op-amp will do the job, 1st stage amplifies the voltage by x100 (For example) as it's hard to compare against millivolts, the second one compares the high level output against a reference (Like 2.5V or whatever), toggling the output when surpassing it.
Add a small hysteresis so it doesn't toggle like crazy (For example, switching at 1A, releasing at 900mA).