You're exactly right. We have a model (Lambda CDM + GR + a few other details). We have ways to generate descriptions of how the universe would look given certain parameters (H0=70, Omega_Lambda=0.73, etc, etc) and we basically just see what range of parameters gives a universe that looks like (quantified using some statistics) the one we see through our telescopes.
But this is just phenomenology. The next step is working out the physics. For example, let's say we know there is X amount of something that looks like a cosmological constant - but what is that. This is what e.g. the search the dark matter particle is about - we know there is something that is cold + collisionless but what particle is it.
> So then you argue that Λ isn't constant. Maybe it depends on time. Or on distance, which I guess just makes it a polynomial. Something like that?
Yup, I'm pretty sure the only models we have tested are wCMD which allows w (the equation of state of DE) to be something other than -1 (which is what the cosmological constant is). There is also w(a) which parameterizes the equation of state of dark energy as a linear function of scale factor (just think of it as time, a=1 now a=0 at the big bang). So linear rather than constant. We haven't gone to higher order than that.
The downside to adding parameters though is that, while you can always fit your data better (or at least as well) with more parameters,
1: Your error bars often blow up
2: Getting from phenomenology to physics might become hard. There are some models people have proposed that might allow us to fit the data, but then you need to explain why w changed in a very particular way at a very particular time. Basically it starts to look a little like overfitting.