Vale of Glamorgan, Wales

I have over the years pruned my Olearia lightly after flowering and I know you can cut out any dead branches in April, but as you can see mine over the years has rather taken over and masking my Kerria Japonica and more. Will it kill the bush if I cut the big branch on the left hand side of the picture, which is the main culprit and spreading too wide, cut that off just above the ground?
I am a bit timmid to do it. If it is ok then I can prune some of the others back gradually year by year not to give it too big a shock. I am worried it is like the ceanothus, cut too hard into old wood means disaster!

• 15 Feb, 2009