Do the bridge saddles have to move 3/8"?
I make it 3/16"
If your new scale is 3/8 shorter, that means your 12th fret to bridge distance has been shortened by a proportional 3/16", that's not so much.
If you have intonated using a meter, the intonation is probably flat above the 12th anyway and needs to be tempered, which would bring the saddles forward slightly. You might find you can just get away with it. If it doesn't have through-body stringing, there may be enough room in the pickup cav. to shift the bridge a couple of mm forward as well.
I wish you Stateside guys would go metric it's much easier!