Can I cool an LS swap with a stock-style radiator instead of an expensive LS conversion radiator?

Often yes — that's exactly what Speedway's budget approach is for.

Two differences you have to solve:

1. Upper hose size and side: most LS engines put both hoses on the passenger side, and the upper hose is smaller than on older engines. Speedway: LS upper hose is 1-1/4 in (smaller than SBC/BBC) (ICT Billet says the upper hose connection on all LS pumps is 1-3/8 in; verify).
2. Steam line: On most LS engines the thermostat sits low, on the water pump, instead of up top like older small- and big-blocks. GM added steam lines in the cylinder heads to vent trapped air and keep hot spots from forming at the top of the engine.

The budget fix: Speedway's LS swap radiator hose adapter (listed at $24.99 in its article) fixes both problems at once: a 1/4 in hose barb takes the steam line, and it adapts the LS water pump's 1-1/4 in upper outlet to the 1-1/2 in inlet on a typical radiator. From there you can keep a radiator you already have (you may need some clever hose routing across the radiator), or use a universal double-pass radiator, which puts the upper outlet on the correct side and can cool better than a single-pass. Speedway notes universal radiators will still likely need some modification to fit.

The easy (pricier) route: a direct-fit LS radiator from makers like Afco, DeWitts or Frostbite. It drops in and hooks up like a stock part.