Thanks for taking the time to document and post your single-basket experience!
I also did not notice too much difference in terms of taste between the Strada/precision basket and Tidaka/normal basket. Are you using a bottomless portafilter? I noticed that the LM Strada basket has a tiny bit more tendency to channel, but it's really a tiny difference we're talking about here.
Re your Tidaka basket popping out, it's almost certainly due to the tight portafilter spring. If you want to use the Tidaka basket, you can adjust it by taking the spring out and bending/rounding it slightly.
I know your goal is to pull 7g shot....
For singles, I noticed that 1g dose difference is enough to affect the taste significantly (compared to double basket). So if you have the chance, I highly recommend to updose to 8.5g (coarsen the grind a bit), and cut the shot when it blonds (around 18-22g gram shot weight for me), with around 25-32s shot time. That can give you thicker body, more flavours and still maintain the balance. But considering we have different equipment/roast, it's possible YMMV.