Unlike gain, stability is a DC-to-daylight phenomenon.
Meaning...the LPF presents the "Right" impedance to the amp output in
the passband, but consider: the LPF is a gross mismatch in the
stopband. Evidently the output of the amp is only conditionally
stable above the LPF cutoff, and the stopband impedance of the LPF
lies in the unstable region.
This kind of thing can be quite common when not considering stability
across all possible frequencies that the amp has a gain1.