Chris, thanks. The WPEngine folks claim they’ve scoured the Apache/PHP logs. The theme/plugin incompatibility possibility is one we haven’t yet systematically explored as we have lot. And, since the problem is intermittent, the sequence of steps is nothing special: we just bring up the page and sometimes the mashup shows, at other times it says “Loading…” and never loads.
Oh well, I’ll keep in touch if further clues.