It sounds like your model is of this form; $$Y_i|x_i = f(x_i, \beta) + \epsilon_i,$$ where $Y_i$ denotes the $i$th measured outcome, $x_i$ is a vector of covariates for that outcome (i.e. experimental circumstances), which with (unknown) parameters $\beta$ determines the expected value $f(x_i, \beta)$ for that observation. The $\epsilon_i$ are the error terms, which describe everything that affects $Y_i$ not captured by $f(x_i, \beta)$ - i.e. experimental errors.
Before getting into analysis, it's always good to ask "why do you want to do this analysis?". The answer to this question determines how much you should worry about Normality, or whether a transformation is needed. Suppose, as is common, you want inference on the value of $\beta$. If you believe that $f(x_i, \beta)$ captures the mean value of $Y_i$ correctly, and you believe that $Var(\epsilon_i)$ is the same for every measurement, then classical linear regression can be used for inference about the value of $\beta$. Despite what many textbooks advise, you do not need Normality here; in reasonable sample sizes your confidence intervals and tests will be almost perfectly-accurately calibrated.
If you still want inference, but don't believe the constant variance, use robust standard error estimates. If you don't believe the mean follows $f(x_i, \beta)$ or that the variance is constant, robust standard error estimates still give you accurate inference on the best-fitting line of the form $f(x_i, \beta)$, where "best-fitting" means "least-squares". And if you don't believe the mean follows $f(x_i, \beta)$, or that the best-fitting line of this form is a useful thing to know, you can always fit a more flexible mean - spline representations of covariates $x_i$ are a good way to do this. Absolutely none of the methods listed require Normality - or transformations of the $Y_i$.
So when do we require Normality? If you want to do predictions, of new $Y_i$, for most methods you'll need a model (though it need not assume Normality). If you want to compare models, well, you'll need some models, but that's a tautology. If you have a tiny sample size, doing model-based inference on $\beta$ may be the only viable approach - but then you'd likely have no way of assessing whether your assumption of Normality (or whatever you assumed) was reasonable.
When do we need Box-Cox? If we have little idea about the form of $f(x_i, \beta)$, but believe that errors around $f(x_i, \beta)$ "should" be Normal, then Box-Cox may help find a better form for $f(x_i, \beta)$. But it relies on there being underlying Normality, at the "right" model, and this is hard to justify in many situations.
In short, rather than deal with hard-to-justify transformations, there is a lot you can do with just a mean model. If the original units of measurement help you (and your colleagues) think about what the data tells them, I recommend hanging on to those units, if possible.
