Since type II supernovae are hydrogen-rich and type I are hydrogen-poor, would shooting through a hydrogen alpha filter be a quick-and-dirty way to distinguish them without spectroscopy?
Hydrogen present (in type II) or absent (in type Ia,b,c) in the spectrum would be a better description. The problem is other events also pretend to be type II supernovae like novae, dwarf novae, luminous blue variable supernova impostors etc. You need the detail of a spectrum to be sure of what you are looking at. The spectrum of supernovae also vary with time and with type II, H alpha emission does not really dominate the spectrum for a lot of it and when it does it is often a combination of emission and absorption (a P Cygni shape line). When typically measured a few days after discovery the type II spectrum can be almost devoid of features. This one was discovered early so did show some H alpha emission early on but even then the dominant feature was that is was very blue and currently it looks almost like an A type star but without hydrogen absorption lines, just a very small P Cygni H alpha line. See here the evolution of the spectrum over the first week.
https://britastro.org/observations/observation.php?id=20230528_140227_583fe05370c08912
For a more general overview of using spectroscopy to classify supernovae I did a talk about it here
https://britastro.org/videos/using-low-resolution-spectroscopy-to-confirm-supernova-discoveries-2
In essence it is done by matching the spectrum to templates of various types of supernovae taken at different times
Attached is an example from the presentation of type Ia and type IIP spectra around or after maximum light. I think it would not be straightforward to distinguish them from just H alpha and broad band images
Cheers
Robin
