Those intersections and bus lanes are too dangerous, in my opinion, to be useful for cyclists. Buses can muscle their way through and will come out on the 'alive' side of any conflict. Cyclists are always taking a large risk going through such a nasty junction. If I were cycling that, I'd avoid the bus lane and take the general traffic lane through there (getting out of that left turn lane as soon as I could) - it's much simpler and drivers will know instantly what your intentions are. Merging right into those bus lanes from the left turn lane, it's just asking for trouble, because what you must do in order to get where you're going ends up going against what the road markings say you're doing. This is going to confuse every driver who sees it.
I know the road markings say you 'can' do what you did, even though technically you're cutting across a lane of traffic. Really that point at which you had the conflict should have a triangular give way marking, making you give way to that lane of traffic - but it doesn't. But I wouldn't take that route if you paid me a million dollars. Well, okay, maybe a million (a million can pay for a lot of surgery and physical therapy, and could give my surviving relatives a nice nest-egg if I got killed), but I'd refuse a few thousand for sure.