Regression vs Classification for a problem that could be solved by both
Without having the data and running classification or regression, a comparison would be hard because of the metric you use for each family is different. 
For example, comparing RMSE of a regression with F1 score (or accuracy) of a classification problem would be apple to orange comparison.
It would be ideal if you can train a good regression model (low RMSE) because that would give you information more than the original pass/fail question. From my past experiences with industrial customers, 
First, train all 3 models you have mentioned and then present the outcome to your customer and let them give you more direction on which models/outputs are more meaningful for them.
I have a problem that I have been treating as a classification problem. I am trying to predict whether a machine will pass or fail a particular test based on a number of input features. 
What I am really interested in is actually whether a new machine is predicted to pass or fail the test. It can pass or fail the test by having certain signatures (such as speed, vibration etc) go out of range.
Therefore, I could either:
1) Treat it as a pure regression problem; try to predict the actual values of speed, vibration etc
2) Treat it as a pure classification problem; for each observation, feed in whether it passed or failed on the labels, and try to predict this in the tool I am making
3) Treat it as a pseudo problem; where I predict the actual value, and come up with some measure of how confident I am that it is a pass or fail based on distance from the threshold of pass/fail
To be clear; I am working on a real problem. I am not interested in getting a super precise prediction of a certain value, just whether a machine is predicted to pass or fail (and bonus extension; how likely that it is to be true).
I have been working with classification model as I only have a couple hundred observations and some previous research showed that this might be the best way to treat the problem. However I am wondering now whether this is the right thing to do.
What would you do!?
Many thanks.