Most controllers charge only one bank of batteries. I have the Morningstar Duo which charges both the coach and chassis batteries. I have it set to put 90% charge to coach and 10% to chassis. If the coach batteries are fully charged, it will automatically route full charge to the chassis, and vice versa. I have one Kyocera 135w panel for two golf cart batteries, and it does a pretty good job of keeping them up even while boondocking. This is way overkill for your purposes, but the Morningstar Duo is one excellent option to keep both banks charged with a lesser solar panel. It would also give you the option of adding significantly more solar in the future, if you desire.
With a solar panel and a Duo controller for it, you have no need for that Trick-L-Start charger. Some people will tell you that a 10w solar panel is worthless, but that's simply not correct. However, if I were going to rely on a solar panel to do nothing but keep batteries up while in storage, I'd probably get a 40w and live comfortably with the overkill.