This maybe a long shot for the E53 but the E60 has a vacuum operated a exhaust valve near the tailpipes that open up under increased vacuum when engine revs over a certain rpms maybe that valve is stuck open OR you forgot to plug in the vacuum hose for it after your repair.
All this is me assuming the E53 has the same setup there as the E60's.
Hope this helps.
I stumbled on this thread as I have intermittent opening of that valve when driving thru heavy rain......??

__________________
E39 528i 1998, 5 spd black, 350Kkms and going..with new owner.
E60 545i 2004, 6 spd steptr. mystic blue, Tom Sellick's old car, Adaptive steering, Adaptive Xenons, Anti dazzle mirrors and FAST!!!, BT&Aux+SB Chime+Dig speedo+iDrive discl. Coded