There are probably more qualified persons than me on this forum to answer your question but I guess in the twotters case, doing a battery start might not produce a reduction of ITT to enable them to have a big enough buffer to max ITT during the start sequence. Having the power from a generator running on the right engine will likely produce more juice for the starter so that it can motor down the ITT and therby produce a bigger buffer to max ITT.
The temperature of the turbine (I'm looking at turboprops now, pure turbojets and turbofans may differ) varies very much with the amount of air that goes through the engine. In the initial part of the start sequence when starting a hot engine, the starer is engaged and this will produce a flow of air through the engine. As this ventilates the hot standning air in the turbine section, the ITT (interstage turbine temperature, name may be type specific to the Saab 340) will start to decrease. Different operators may have different temperature limits due differences in engine lease contracts. In our case, the ITT needs to be motored down to 150 degrees C in under 30 seconds. With a poor GPU or bad batteries this might not happen. In that case the start is aborted and the starter allowed to cool before the next attempt. If the limit is made within 30 seconds, we may select the fuel to on, and then we look att the Ng, Fuel Flow and that ITT is rising with normal speed. The ITT will reach it's highest point well before the Ng has established at idle. As the Ng accelerates further, the ITT drops and stabilizes at idle reading (air flowing through the engine increases through out the startsequence up to idle and this counteracts the initial ITT rise).
Selecting fuel on too early will give the ITT a starting point from a higher value and may result in a hotstart as the initial acceleration of air will not be able to counteract the rise in ITT.
/LnS