Tuning down the strings could be a headache. A shortcut is to not tune the strings down. Loosen the claw on the back. A half step shouldn't effect the way the tremolo sits too much. If it does, you need to do a full setup on the tremolo.
The best way to think about a floating trem system is to realise it’s a LITERAL balancing act, like an old fashioned pair of weighing scales.
If you have half a pound on one side, you need half a pound on the other side to make things balance.
If you tune down a half step, you’re reducing the tension on the strings on the fretboard. To make the bridge “balance” you need to reduce the tension on the strings by the same amount. This is done by loosening the springs in the back of the guitar, which in turn is done by loosening the trem claw (the thing screwed into the body of the guitar the springs are connected to) so that the springs are under less tension as they’re now stretched over a slightly shorter distance.
I have a JS1000 that I keep in Eb tuning and once I’d done the setup, it works perfectly. And I mean PERFECTLY