It sounds like you have the protective cover on your RPTV screen. If you're brave enough, you could remove the protective screen and that'll reduce reflections.
I have a cover on mine, however, I'm waiting for my son to be taller than the TV before I remove it.
As far as the blinds go... Depending how they are mounted.. you could probably mount an old fasioned pull down shade behind the blinds inside the frame and that'll block a lot of light and you would hardly notice the roller shade behind the blinds. You can find roller-type shades that can be adjusted width wise for pretty cheap.
Let us know what you figure out.