Should the transmission cooler run through the radiator or separately on an LS swap?

Separately is better for anything performance-oriented or towing. Why a separate cooler: higher-stall converters make more heat at higher rpm, and the cooler in the radiator tank often isn't enough. It also dumps that heat into the engine coolant, so if the engine runs hot, moving to an external cooler helps both. Speedway's 4L60 guide lists a large external cooler as a key upgrade, since heat is what kills that transmission.

Speedway's install ran an aluminum 6-pass cooler mounted low in front of the radiator (in line with the grille opening for airflow), with 3/8 in hose from pre-bent hard lines. Keep the hose away from sharp sheet metal — bulkhead fittings help.