Welcome to the forum! I looked yesterday morning after you posted but didn't see anything that met your criteria...so I just kept quiet.
There certainly might be receivers out there that meet your requirements, but I'm not aware of them (but, I haven't researched it extensively). The challenge, I think, it that you want the internet capabilities that have come into being in the past year or two, combined with a receiver that has at least eight powered channels...all for about $300. There probably aren't many, if any, receivers with eight powered channels, but there are several with seven or nine. You might be able to find a used receiver with the power than you want in that price range, but it'll be more difficult to find one used in that range with both the power and that's new enough to have the internet capabilities that you want. It's easier to find a 7.1 receiver that can be set up as 5.1 plus a set of "B" speakers, but then you don't get the 6.1 experience that you're after.
Have you considered using your Sony for the outdoor speakers? That will lessen the requirements for your new receiver. Also, you could use the Sony simply as an amplifier if you get a receiver that can power six speakers in your main room and has pre-amp outputs for a second zone.
For a network receiver that can handle 6.1 in your main room and that can power two speakers in a zone 2 - but not at the same time - the Onkyo TX-NR626 may be a good place to look. I own that model and have reviewed it
here and on
Amazon. It's full price at $500, but it goes on sale and should start dropping in price between now and summer when then new models come out. It also has pre-amp outputs for zone 2 if you want to use your Sony (or other amp) to power your outdoor speakers. The nice thing about the 626 is that, even though you can't power six speakers in your main room and two outside at the same time, it has speaker connections for all eight speakers. You do have to go into the menu system to toggle between using the second zone speakers or not, though...might be more of a hassle than you want, but maybe not.