Nobody has released an alternative browser engine yet, because of the way the app store works (you'd need specific apps you can only install in the EU next to the worldwide version for instance). I'm sure it'll happen eventually, but it doesn't seem to be a priority for browser makers just yet.

I don't actually think there is official API to check if the device is in Lockdown Mode. But to be clear this is an academic curiosity for now as nobody is actually shipping an alternative browser engine in the EU that is being targeted by a sophisticated attack.