Make sure there is a good ground connection between the RV and the toad. I really doubt the use of a 12 v 'trickle charger.' The so-called trickle charger is just passing the voltage from the RV alternator through to the toad battery. The toad battery should get charged and the battery's internal resistance should limit over charging. Your dead battery points to the fact the voltage isn't getting passed through or is too little to overcome the amperage needs while being towed. I would more likely think there's a bad ground connection. You could test with a meter and a jumper wire from RV ground and toad ground.
Bob & Donna
'98 Gulf Stream Sun Voyager DP being pushed by a '00 Beetle TDI