Mine is doing pretty much the same since last summer.
I think it is related to the fuel pump not priming fuel correctly before cranking the engine on a cold start.
There is a sequence with the keyfob for the alarm system and I think this is causing the problem.
Make sure you can hear the fuel pump well pumping before starting the cold engine.
You can do the procedure twice to make sure the fuel pump worked 2 times and then start your engine and see if the first 10 seconds is idling better.
Mine seem better when doing this.
Otherwise it look like my engine was running on 2 cylinders for the first 5-10 seconds and very low RPM then going back to normal after.
07 Exige S300+,Solar Yellow,Touring Pack, Torque300 tune,RLS Intercooler,Pro Alloy IC tubes,Lotus Stage3 Exhaust,Intake BOE ITG cold airbox,RLS MKII 3.0 Pulley,SaikouMichi Oil catch can,NGK BKR7-EIX Plugs,EV14 525cc injectors,CarbonFiber side scoops+front lip+rear spoiler, Nitron SS Shocks, PC680 Battery,JVC KDNX5000,CompactSub Kenwood KSC-SW1,Tinted windows, Lotus Sport windshield+front plate decals,HID Kit 6000K,Parking Light,license Leds.
2013 Scion FR-S Firestorm
2016 Subaru WRX STI Blue