Oh well... If I loosen the screw on my shifters, the screw and the bracket come out at either side (front/back) so I can directly remove it from the bar without any sliding whatsoever. It's a faff to get them back on but I've managed to do it on multiple occasions... That might be the case with yours too. My experience is with st-r505's and st-r8020's.
I could unscrew the shifters from the bracket, but the brake hose goes through an internal channel through the tops of the bars so they're pretty much stuck in position even without the bracket. I do like to make things difficult for myself...
Oh well... If I loosen the screw on my shifters, the screw and the bracket come out at either side (front/back) so I can directly remove it from the bar without any sliding whatsoever. It's a faff to get them back on but I've managed to do it on multiple occasions... That might be the case with yours too. My experience is with st-r505's and st-r8020's.