Ya without a jtag or RGH (Reset Glitch Hack) system, you can't play XBLA games on a flashed Xbox 360. Need either the jtag or RGH option to do that.
For you the best option would probably be the RGH method since it's newer and more support since it basically took over the jtag systems (though they're still out there).
If you still want Xbox Live, then you may want to look into the DemoN chip by Team-Xecuter, which allows you to have a modded box to play backups and have an RGH Box all in one.
to answer your question as you asked it, yes you can connect to live and download the game, you have nothing to worry about as long as everything was and is being done correctly!
Oh my bad, I didn't read that question right... What peluynati said is perfect, I thought you were downloading the game not from Xbox Live and trying to play it... my bad!
