I have a set of samples that I'm trying to fit a (continuous) distribution to from a list of possible distributions.
The main question you need to deal with is why do that?
What are you ultimately trying to achieve? What does this do for you that you can't get some other way? Why would your data come from some such list?
Is there a "smart" (statistically speaking) method to do this?
Yes: In general, don't. It's going to result in overfitting.
However, if you must do this, you should go to some length to avoid assessing the suitability of the distributional model on the same set (or subset) of data you use to select between the models. (It's also very important you not use that same data for inference involving parameters.)
Sample splitting/cross validation will get you a long way.
For each distribution, find the best parameters using for example MSE.
Why would MSE be a good choice?
For each distribution + best fitted parameters use Kolmogorov–Smirnov test
1) K-S assumes a specified, not fitted distribution.
2) you're presumably comparing distributions with different numbers of parameters; why should they be comparable (models with more parameters should give a better fit -- if this is your criterion you'll simply choose a model with lots of parameters)
3) a test answers the wrong question (the question is whether the model is reasonable for some as-yet-unstated purpose, not whether some hypothesis test rejects or fails to reject)
Does that make any sense?
I think it's ill-advised, but there's not presently enough information to provide a lot of guidance toward something better.
In 2, why not use the MSE as criterion?
First, what are you calculating MSE of? Why is the MSE a good measure of your loss-function?
Why do we need these tests for?
It's not clear what you're asking here. I don't think they serve any useful purpose for your kind of problem. Why would you use them for this?