For *many* single-output converters, without a standby supply, the most likely outcome under short circuit or overload is that the supply will hiccup. The MeanWell one you linked seems to state it has a latching fault protection, so it may have some function built into the controller chip to ensure this.
Normally, hiccup fault is ensured by design because the circuitry on the primary is powered by a winding off the transformer. If the main switching operation stops, that supply will disappear. Supply will switch off, then the bootstrap process will begin again. Result will be a brief output every second or so. Each start up cycle is more or less like a short circuit to the PSU as the output caps are usually discharged, so only primary current will be limited by the short term OCP of the controller until the Vcc cap is emptied. This is quite stressful for the main FET if the period is short, so avoid doing it.
Some supplies have a second independent power supply for the control IC (a good, familiar example is the humble ATX computer PSU - the 5VSTBY transformer almost certainly supplies primary-side Vcc to the rest of the system.) In those cases, the 'standby' rail will behave like a hiccup function (and sometimes, shorting the standby rail will bring the whole system down), whereas the main rail may indeed have a latching fault detect. Some others have a way to keep the controller IC powered all the time from the mains, or to latch this state as long as mains is present.
However, in all of these cases it is not really safe to assume the current limiting is anything other than a last-ditch attempt to save the power supply or its load from destruction. It is very line-voltage dependent, so a particularly high mains voltage will mean a greater output power limit (because it's based on primary-side current: higher mains voltage, less current, for a given power), and it's also not particularly well toleranced, so paralleling supplies and relying on OCP to balance them is very risky business (not saying you're doing this, but I've seen people try.) Running a supply at its power limit, or bouncing right off it, for some time will probably shorten its life. The power limit will probably vary significantly with temperature too.
To do what you want, you do really need a proper charger. Even if you could do some kind of current limiting on the PSU, a proper charger will do things like compensate the charging voltage for temperature, which is mandatory for good lifespan of lead acid batteries. It might even do desulfation cycles for batteries that benefit from that.