The power supply has over-current protection. In most cases that means that exceeding 30A momentarily will cause the supply to shut down until the load is removed (in some designs you have to cycle the power switch to reset the supply). If you really want to recharge a low (say 50% or more depleted) battery at the same time you are transmitting then you really need a larger supply. If you are okay with not transmitting until the battery is nearly fully charged then the 30A supply should work okay. Once the battery is fully charged it doesn't take much current to keep it topped off.
Note also that continually connecting a nearly depleted battery to a 30A supply without some sort of current limiting on the battery may shorten the service life of the battery. For example, if the battery is down to 10V and you connect it directly across a 14V power supply the battery may draw more current than the mfgs specified maximum charge rate for a period of time.
Very simple solution here even though I doubt that power supply will ever be over drawn. Simply get a 1 farad carcaudio electrolytic capacitor and place it in parallel with load. It will easily feed rig during SSB peaks and reduce peak demands on power supply during charge. It will also stabilize voltage when just on batteries. Audio buffs discovered this trick years ago because it WORKS! For some reason some are against usingna proven setup.