I think the answer to the original question is not necessarily dependant on peak power. My chipped S3 managed to keep up with (but no where near faster than) Glen's C4S, but that was down to very tight twisty roads, good low end torque in the S3, and a good chassis and suspension set-up. As soon as the road straightened up I didn't stand a chance.
The ability of an S3 to keep up with an STi, M3 or RS4 in any corners would depend more on suspension and the roads you're driving on rather than outright power. S3's a good on tight twisty stuff, but you need to look at the suspension, ARBs, polybushes, geometry, and possibly the haldex controller to make it a more equal fight.
On a straight, the S3 won't get going as quick as the rest but would probably take the scooby from 40-100mph. You'd be neck and neck with the M3 till you got to silly speeds (120mph+) then the M3 would pull away. RS4 - forget it. You will also be changing gear a lot more than anyone else.
I run mine at 0.9bar when i'm just running around town. It should give me about 240bhp and that's dire compared to the full boost shove. Much faster.
One thing, 350bhp is a little on the high side for some of the internals. Most respectable tuners (AMD, MTM, Sportec, Star, etc) keep the levels down to 330bhp for piece of mind. Don't be surprised if yours is making that when you get it.