Bleaching polyester will destroy it, and there is no way to lighten it.
I'm sorry, you just can't do it, it would be best to find a different coat.
You could try to compensate for the brown dye that's in it already, but depending on the purple you want, it just might not work out.
Also, you have to consider that the poly and cotton dye differently. You need to hit it with a poly-dye (iDye Poly) and a dye that will work with the cotton. This means not only correcting for brown once and getting it right, but doing it twice and getting THOSE colours to match.
This will only possibly work for a muted, muddy, reddish brown kind of purple.
Typically, you want to use a dye remover, but they are already at risk and likely to end up splotchy even on a 100% cotton fabric. With the high poly content, it's just not going to work. It WILL be uneven and splotchy or just not remove at all.
Bleach will destroy it. Bleach weakens even natural fibres, like the cotton, and it's not going to lighten the polyester at all. It would tint it yellow if the polyester was stark white, but it will not lighten it to yellow, it will do nothing (this is just like wig fibre), and it will destroy both fibres if you try to get it to.
Actually in the pipes:
- Remaking Ser Loras doublet.
- Luke Skywalker, A New Hope
- Wind Waker starter costume in swim material
Eventually: Paul from Dune, Faramir, Harry Sullivan