The motor will almost certainly be fine with a 11.1 V lipo, they are usually designed for 12V or a little more. However the trigger contacts will see arcing with a 11.1V lipo and this will degrade them much faster than with the usual 7.4 or 9.6 V nimh. So while with a 7.4V a mosfet is somewhat optional its definitely not optional on an 11.1V lipo. If you don't want to rewire the gearbox and such you could try an inline mosfet like the burst wizard. But really the best way is to wire in a BTC or other mosfet into the trigger and the external part so that the large voltage doesn't run through the trigger contacts. In addition it would be a good idea to switch to deans connectors for 11.1V to reduce the temperature/resistance of the connectors. Tamiya connectors do get hot when dealing with high voltage and current batteries and its possible to melt them on full auto so its usually best to switch to a low resistance connector as well.
There are a few different ways to reduce the firing time depending on the root cause of the slow speed. Some of the problem could for example just be a long trigger pull and reducing that down with a bit of shimming of the trigger could make a pretty big difference. Then it could also just be the motor struggling to turn the gears and pull the spring, a softer spring and some shimming of the gears could help reduce the issue. You could also move towards a torque motor that will have a moderate ROF but get started really easily. Or you could have a lot of electronic resistance and want to rewire your gun and switch to deans. 11.1V is kind of a hammer approach, it will help reduce the problem but it doesn't focus on the core issue causing the slow ramp of the motor, it just pumps more juice at it to get it moving faster without attacking the reason why its slow, but in the process it puts more strain on everything in the guns electronic system.