The fake fire type I have seen the heating element is buried inside the bloody thing deeper than any oil heat I have seen. Based on that line of thinking the fire place one is much safer as the case does not get hot and only one vent does and it is just air and unlikely to ever burn an animal as it never gets hot enough to. Oil heaters typically have exposed fins than get quite hot.
The issue I see is how much heat is delivered and will it dissipate too fast. I tried a forced air system Ceramic and it worked ok not great for my situation but then again the oil heaters also I had issues with. No situation is the same, one solution may not work for all situations. Trial and error set it up in a room and try it. You maybe able to get a Constar room temp out of it with little or no spikes depending where you direct the vents. I was unable to get mine to work because the poor insulation my place has (seaweed in bags) Same reason why oil heaters didn't work well.