If we forget about the inverting op-amp and just consider KCL.

Why does that particular route of analysis give me +4.5V?

You can't just "forget about the inverting op-amp". The op-amp, acting as a voltage-controlled voltage source, connected between the op-amp output terminal and the ground node (your diagram does not explicitly indicate this) is providing current to the 96 kΩ load, as well as (it turns out) an identical current through the 96 kΩ feedback resistor. The ratio of the 96 kΩ feedback resistance to the 32 kΩ input resistance sets the op-amp circuit gain to -96/32 = A = -3. Since Vo = A Va, and Va = 1.5 volts, then Vo = -4.5 V.

Your KCL "analysis" at the ground node does not account for all the currents at the ground node because it ignores the implicit ground node current returning to the op-amp represented as a voltage-controlled volage source.

I understand that +1.5V fed into the inverting op-amp will give an output of -4.5V. But using KCL at the ground node give me +4.5V, why is this?

Because you didn't apply KCL correctly to account for ALL the current at the ground node. It was just a fortuitous choice of component values that made it appear that Vo was a function of Va with the wrong sign.

The op-amp can be modeled as a voltage-controlled voltage source, in which case its output is represented by a voltage source, Vo, connected between the op-amp output and the ground node. Note that this voltage source, Vo, is conspicuously missing from your diagram. The current through the 96 kΩ load resistor, as well as the current supplied to the 96 kΩ feedback resistor, must flow back to this voltage-controlled voltage source which has one terminal connected to the ground node. You do not present an equation that accounts for this ground return current.

The output Vo should not be a function of the load resistance. Try substituting 100 Ω, 1 kΩ, and 10 kΩ load resistors in your "KCL ground node calculation" to find out what you get for Vo with those three different values of load resistance.