The discharge rating only indicates how quickly the battery can provide power to the electrical system. A higher discharge rating won't overwhelm the car. However, batteries with higher discharge ratings WILL cost more.
To answer the OP question: The AMP MT and DB do NOT have an ESC that has built in low voltage protection. To safely run a LiPo battery in the AMP you will need to buy a low voltage alarm. These are inexpensive little gizmos that connect to the balance lead on the LiPo battery and will sound a LOUD alarm once the battery drops below a preset voltage. Once you hear the alarm, stop driving and recharge the battery. I have the
Venom "Low Voltage Monitor"- and it works quite well and can be configured for various voltage levels. You can find cheaper LVM devices on Ebay - for just a few bucks - but I can't speak to the quality of those devices.
LiPo batteries, unlike NiMh or NiCd, do NOT respond well to discharging below a certain point. Discharging a LiPo below 2.8V per cell can RUIN the battery! You can run a NiMh completely flat, but not a LiPo! There is some debate about what the lowest safe discharge voltage is for a LiPo, but most agree that 3.2V or 3.4V is a the best, safest cut off point. In fact, in practical application, you'll notice a distinct drop off in performance about 3.7V so you'll want to stop and recharge well before battery damage anyway.
I have two Venom 7.4V 4000mAh batteries. I can run my AMP full throttle for about 40 minutes and the battery will have only discharged to about 3.8V. And, unlike a NiMh, LiPos typically don't get so hot during discharge (maybe 90 deg F) so you can put them back on a charger immediately. A NiMh gets hot and needs to cool down between charge discharge cycles.