^^^ It is true that you (usually) need more than 12V to charge a "12V" battery. But that's at the battery posts. Not at the power source. DC-DC converters can certainly generate 14V from less than 12V.
^ Whereas it may be fine to use 2 charge controllers to the same battery off one panel (if they are set up properly), it is more problematic to use 2 MPPT charge controllers with separate batteries, as their respective algorithms for MPPTracking may not be compatible and things may go wonky somewhere. That would depend on their respective specs. (But I would think that having one of them be PWM -shudder- instead might help with that.)
OP, as to ways to do what you are trying to do, there are many. Inverters, charge controllers of various kinds, converters, isolators, a second panel, etc. But the "best" way would be a function of what you already have equipmentwise, your sensitivity to the cost of anything new, and losses / efficiency you can live with. IOW, it's a very wide question with no obvious answer.
One complication to keep in mind is that you likely want some sort of smart control of the main battery voltage, with floating capability etc. In addition to the fact that you are using 2 different chemistries in your 2 batteries if your starting battery is lead acid. So some smart device would need to be involved.
Having said all that, if I had to take a guess at the cheapest / easiest way to go, I would assume that you already have a 110V smart charger, and just using a cheap inverter (can be very small but should be decent quality, preferably pure sine) would work just fine then, well at least if you have enough solar power to overcome the losses (I'm guessing a continuous 5W or so for the inverter for the sort of power needed). You could be set for well under $50 if you don't already have an inverter. Is it elegant? No. Would I do that? Yes if in a pinch, no otherwise cuz I think I could do better and I'm a nerd for stuff like that. A cheap alternative to the above might be that one PWM controller off the solar panel directly to the main battery but they are not very efficient, so meh. Or, yes, something like that Tekonsha but you'd want to look carefully at the specs, especially regarding the feedback issue. Me, I'd go with something like a Victron Orion DC-DC charger but that's significantly more moolah.
Now, as to whether it'd be better to have the panel connected to the starting battery instead, that's another whole can o' worms.
