Is this a sensible approach for a 3x3 factorial experiment?

Hi all

I'm hoping that someone can let me know whether this is a suitable approach for analysing a dataset.

I have a 3x3 factorial design, so 2 treatments with 3 levels. From each of the 9 treatment combinations I took 4 replicate samples and measured several variables, just once. I am interested to know what explains the response of one of the measured variables in particular.

Firstly, I thought to do a 2-way ANOVA to test the effect of the 2 treatments on this variable.

But I'd also like to know whether the other measured variables might explain some of the response of the main variable, so I thought to do a linear model...

As there are no blocks or plots or repeated measures I assume I cannot do a linear mixed model as I cannot define a random effect?

I thought to do a general linear model instead, including all variables until the model with the lowest AIC criterion is found.

Does this seam like a sensible approach?

Many thanks!