Is it possible for an intermittent load to draw too much current from a charger that is also connected to a SLA battery?
Your intermittent load is 12A. This is C/8, well within the capability of a 100A battery to supply without breaking sweat. For 5 minutes, it's only 1Ah.

As long as your load can tolerate the change of voltage between 13.8v (or whatever the charger floats the battery at) and 12.something (the voltage when the battery is supplying the bulk of the load) then the load will be supplied no problem.

A dumb charger will have no problems, it will supply a safe limited current into whatever battery voltage is presented to it (as long as it's within normal limits). If it's claimed to output 36W max, then that's about 3A when the battery is loaded. When the battery reaches float voltage, it will take whatever minimal current it needs to.

It's possible (but unlikely) that your charger might be 'smart' enough to get confused by the change of battery voltage that happens intermittently. It would be a pity if it deduced that there was something wrong with it and tried to shut down. It depends who wrote the firmware for it. Check it behaves itself under your loading regime.