Thanks for weighing-in linda, WME, and payaso.
The coach battery does measure a 12.4v charge with a meter.
When the pigtail is connected to shore power there is an audible "click/chunk."
Indeed -- as was suggested, the issue may be fuse-related. When pulling them I noticed that one, and maybe two, are shorted-out (novice mistake for not checking them initially!). I'll get new fuses swapped in and will update here. It'd be nice were the solution such a simple fix. If that's not it I'll continue as suggested -- disconnect the battery and clean the relay contacts with an emery cloth, etc.