[JRTJH]

Every response has essentially said that your Pilot is not suitable for towing a trailer in that size/weight range. So........ Adding a transmission cooler won't change that. Again, it's up to you, but the overwhelming response is, "Don't do it". As one who has "never towed a trailer before" you might do well to listen to those with experience. There's no need to learn by mistakes in this situation.