yeah, i dont really get the premise tbh. The stock engine has a variable vane turbo and hits full boost extremely quickly already. Sure, it doesnt make much below 1500rpm, but why are you ever trying to request full power that far down the rev range? Select the correct gear and your at 2500-3000rpm and response is instant. Its also modifying something where the solution already exists, and in the case of the E90 that solution is the 330d, or 335d for even more go.
One thing the bigger engine does, is makes the lower RPM range more flexible, simply by virtue of 50% more capacity and thus 50% more power even without any boost. If you do find yourself in the wrong gear it doesnt matter as much.
Maybe its just an age thing. I remember when i was younger, modifying cars, and always wanting to "do something different".... I did daft engine swaps that looking back now i can see made zero sense. My brother had a 320d, and we did various bits to it including injector upgrades and larger turbo etc. Most folks on the BMW forum pointed out we should have just bought a 330d. Somehow we thought we knew better... Some time later i bought myself a 330d, and immediately realised that indeed they were right. It was just better in every possible way. Not only did it make more power in a much nicer way, it had a wider power band, it was smoother, it sounded nicer etc etc. A completely stock 330d was better than that tuned 320d ever could be, and ofcourse, the 330d could easily be remapped for even more. He eventually stripped the 320d for parts because he couldnt sell it for what it was "worth" and having learned theres no replacement for displacement, bought a 130i.
Eventually i realised the common approach that i rubbished when i was younger was common because it worked and people had gotten to that point by iterating thru the other options and finding the best one.
Given how difficult its been for OEM's to get electrical turbocharging working and viable (they've been working on it for at least 20 years) I think the idea that your going to bolt such a system to an existing engine, and somehow have it work properly, is just nuts. Especially when the intention is only to provide a tiny bit of boost below 1500rpm, where you basically never should be anyway... Maybe you find the process of building the thing fun, but again, with time i've realised that jumping into something that Just Works has its benefits. If the thing you want doesnt exist, then sure, build it. But in this case, you could either fit the larger engine, or if for some strange reason you want to keep the nasty 4 cylinder, grab the twin turbo arrangement from the 123d, and get that working on your E90. At least someone with millions of pounds has engineered a solution that does actually work, and you just have to tackle the challenges of making it work with your particular car.