@Genista Our sincere apologies for not responding to this sooner. It must have been missed amongst the increase in forum activity during the initial release of the 146.
As you correctly assumed, the engine behaviour during startup in the 146 is currently controlled by MSFS, and we only have very limited (if any) control over the engine behaviour until after the engine has reached ground idle after engine start.
To add a TGT peak and to simulate hot-starting would require a custom-built engine model, which is a little outside of the scope of the project as it stands. Improved engine simulation is something we would like to eventually add to all of our aircraft, so this may be something we include at release in one of our future aircraft, and then back-port it to the 146.
We do have this feedback logged on our internal bugs/feature request tracker so it is something we will consider during development of our future products.
Mark - Just Flight