Yes, describing what a smart battery controller does compared to a dumb controller, as a possible explanation of why your charger only does 13.8v charging.
Probably not a good idea to have both supplying current at the same time, but if both were connected at the same time they would both be paying attention to the state of charge of the battery.
*more confusing technical stuff*
If you did connect and power on both the charger would lift the battery to 13.8v and the PWM controller would probably be happy with it doing that, then the PWM controller would sense that it wasn't providing much current to the battery and change to its absorption charging mode and pull the voltage up higher. That would cause the current from the dumb controller to fall significantly, not that it would care, and the PWM controller to take up the slack, again not a problem. After the PWM controller is happy that the battery is fully charged it would go back down to 13.8v for float charging and then the charger and the PWM controller would share the load.
There's a million and one smart lead acid chargers out there. I'm in 240v land and I think you are in 110v land, so any charger I'm familiar with won't be any good for you. Maybe someone else can give you advice there, but there's a wide selection on ebay and amazon.
They charge differently. A lifepo4 doesn't have the early current tail off that a lead acid does so can use a simpler charging strategy but given how cheap smart chargers are these days that's largely not a concern anymore. You can abuse both lifepo4 and lead acid to get very high charge rates and both will suffer some degree of performance degradation as a result.
lifepo4's big advantages are weight vs energy storage, number of charge / discharge cycles, and not needing to be religiously bought up to full charge. Lead acid batteries need to be regularly fully charged and held there when ever possible otherwise they'll have reduced lifespans.
No need to apologise, asking questions and having them answered is the whole point of this forum.
I'm sure someone will step in and make corrections if I've made any errors.