The stock transmission lines on muscle-era Chrysler products was 5/16", which I agree are a bit small for what you're attempting. The oil-cooler kit I bought for my non-E-body turbo project has hoses literally twice that size, but it replaces the oil filter with an adapter plate--literally all the engine oil goes through it before reaching the engine. Obviously want as much volume as possible in that situation.
It might be possible to route some of the oil through the radiator cooler, as sort of a bypass, but I can't think of a good take-off point to do so. The oil sender port seems an obvious answer, but left unrestricted there might be so much flow you'd potentially do the rest of you engine a real disservice. With some sort of flow restriction, the volume actually being cooled might not be enough to justify the effort. You'd almost have to route the outlet back to the pan; a return to a pressurized location would just result in no flow.
Another possibility would be coming off the back of a 90° filter adapter on the high side, then returning at the low side. The obvious downside to that is you've created a filter bypass allowing cool, but unfiltered, oil into the engine.
You're on the right track, though: A liquid-to-liquid heat exchanger is a ton more effective than liquid-to-air, and a sneaky, built-in one like that would be ideal.
One thing to consider: You don't want your oil at the same temperature as your coolant. There are recommendations regarding temperatures at which oil performs its best. I found the information years back, but if memory serves oil temps should be kept between about 225°-265°F. Too cold and you lose flow, as well as creating additional drag via pumping losses. Too hot and obviously the oil will break down more quickly. Based on the information I found, I decided to forego an oil cooler on both my drag-toy Valiant and my street-oriented Challenger. It did not change the plan to oil-cool the turbo engine. Compressors make a lot of heat.