Wow, there's a lot of skewed 'facts' in this thread!
Variables - Best power option - stroke/cam an LS3 = to 408+ ci = BLOODY expensive!!
And honestly why the hell would you even concern yourself about 411ci or 423 or 427ci capacity? Unless you're ripping out the engine - no one can tell! so don't get wrapped around the axles on that one. If this is a drag car - lucky because it will chew fuel like a demon, that's what strokers are famous for (and making bulk torque)
Stroking an LS1 is not really an economical or reliable option - once the bore liners are removed and the block re-bored - the core is likely to shift and you'll get ovular bores....98% of workshops wouldn't even bother doing it. The later L98 and LS2, LS3 are better for stroking, but funnily enough cost twice(+) as much as an LS1. Getting 380rwkw out of an LS1 - you're wasting your time and money - because within the limitations you have it's not possible whilst remaining legal and Reliable...again assuming you can't go forced induction. (NOS would be a good idea - but wont win any fans from the Police....)
Unfortunately stroking and all the other costs like big fuel systems etc will be 15-20k+++
So if your budget is 10k - only forced induction will be able to get you there and because you've discounted that for legality reasons (legailty?? - honestly pretty well everything about the whole engine and the exhaust you put on it wont be legal anyway - An engineer can sign off on alot of stuff (costs $$$ too) but your twin three inch exhaust with high flow cats will never ever be legal for sound, nor will it pass any emissions test....ever.
So really you have three options :
1. You make a drag car that's not ever going to see the street - CHEAPER way to make power - because you're not limited to "silly rules' and it's safer for your license and hip pocket. (hell, just add a massive cam to a crate L98, crap load of NOS - cheap - thanks for coming. Or adding twin turbos to an LS1 etc.
2. You get most of the car engineered, but can't get other bits certified - it will be illegal though and you just run the risk and play russian roulette with the police and in this case it will be the equivalent of having 6 bullets in the magazine with one or two spare. And also for a street car it will be terrible and you wouldn't even bother driving if you could afford to keep fuelling it and feeding it tyres.
3. You try to stick to the compromises between legality, power and 1/4 mile time and spend an absolute truckload of money to achieve it. maybe?
I'm surprised no one said the obvious - 380rwkw? who cares - remove 100-200+kg from the car and you wont need so much power! Stripping it out for drag times ...then you have legal problems again though......
Overall, doom and gloom aside - the compromises you have made on yourself have made it close to impossible. And also your budget needs to be sorted out.