I believe it's prudent to put the greater portion of budget into the speakers, as you intend, and for you that's seems to be about 3/4. Good on you.
Problem with Lsi series is that they need more amplification than a $500 price point is going to get you. I suppose the only way to get proper amplification, with this budget for Lsi, is buying a very outdated receiver with preouts, and adding a used pro amp or something like that.
The problem with the unknown subwoofer budget is that very different budgets can be called for depending on room volume, and/or apartment living.
I think a good price point value is the SVS PB12, but I know there are other competing candidates from other ID brands like Axiom, Hsu, Outlaw, Epik, eD, and what not. The best value is building your own. There is a good DIY Kappa Perfect build at this forum.
I've never heard them, and they would almost blow your budget by themselves, but I'd be very interested in the Salk Song Towers for $1,700.