RC snubber on bridge rectifier (capacitor rating)
Simply put, Y capacitors are intended for filtering common mode between line and ground/PE.  X capacitors are intended for filtering across the line: L1-L2 or L-N.  This is described in IEC60384-14 which is harmonised with UL 60384 and EN 60384.

You clearly do not have a Y capacitor application if you're not connecting to chassis/ground/PE.

The peak pulse voltage you would expect to encounter dictates whether you would use an X1 or X2.  For up to 2500V, you can use X2; above that and I believe specified up to 4kV you would need X1.

You mentioned that you plan to obtain UL and CE-Mark for the end product.  I suggest you work with your lab or certifying agent on this detail since it doesn't sound like you are using the capacitor in its intended application (line to line).