Just using an aftermarket oil cooler by itself would result in your oil running too cold on most days, for a street driven application. You'd never boil off the water vapor. Using one with a thermostat would probably be fine, and is most likely the best setup for racing as you can have as much cooling capacity as you want with a system like that.

The transmission cooler isn't designed for the pressure. The metal tube would probably withstand it (we're only talking 80psi here) but you'd have to design fittings to connect the oil pipes to it, because the fittings that it comes with are just rubber hoses with hose clamps on 'em. Not good for anywhere near 80psi. Plus, my understanding is that it runs at the bottom of the radiator and will probably run you a bit too cold. But, it could be done. I sure wouldn't do it, but it could be done.

The stock 1800 oil cooler is probably what you want for any non-racing application. It's a simple heat exchanger between oil and water, which means it always moves the oil temp in the right direction, keeping it from being either too cool or too hot. Perfect for street use. Not a huge thermal transfer capability, but it does no harm which is really a pretty good place to start.