Theres a few things going on with this type of setup, I've never gone as far as scoping each pin or trying to get the JOBD output working to see ignition timing or other values to try and diagnose it fully but I've been able to confirm the following on a few cars.
a) Normally with auto shifter it gets signals sent to auto ECU for P/N/D/R/L/2/3/Overdrive/Manual etc . With a manual none of these signals other than NSW (Neutral) is still recieved by the ECU, so it seems auto ECU always considers the car to be in Neutral, as NSW pin is always grounded on a manual box. The idle air control valve & ignition timing values around idle change based on weather you are in neutral or drive with an auto, as being in drive will load up the engine more. This is what I think causes the main issue with stalling with auto ECU's. A dirty hack that can get around this is to bump up the throttle set screw a bit to allow more air to enter the engine through the throttle. Don't forget to recalibrate the TPS sensor afterwards to keep ECU happy.
b) Back to back on the dyno i've tested auto and manual ECU's on a manual converted TT car and the manual ECU made more power everywhere and the turbo transition happened sooner. Presumably auto ecu being in Neutral has less timing compared to if it was in drive and more than likely has different turbo VSV activation maps based on RPM/Speed/TPS/Gear Mode etc.
c) Not all but some manual converted cars on auto have shit top end power, graph on dyno is horrible. Seems likely its pulling timing for gear shift compared to a manual ECU that will just keep going to the limiter. RPM limits can also vary by ~500rpm at times as well sometimes changing for no reason.
You can of course still drive the car with auto ECU and manual box and it will feel ok, but theres more left on the table usually by going to a manual ECU. All of the above is mainly tested with Jspec pre facelift cars. Can't remember if i've seen this behavior with UK cars before or not, but as UK ECU's are way more sensitive compared to Jspec it wouldnt suprise me if it had loads of warning lights and codes. Facelift auto ECU's really dont work well manual swaps as they are lacking certain speed signals so the car hits a rev limiter really early on.
So yeah with all the above mentioned, my recomenation is always manual ecu when doing gearbox swap or go to a standalone ECU and save yourself the aggro.