If a player of mine was having a major issue with perceived capability of their War cleric, and I decided that it needed a buff, then the aforementioned martial Weapon proficiency or allowing War Priest to activate on a spell cast action would seem to cover the major bases.
I think that's the crux of the issue.
At least I am approaching this discussion from a theoretical design angle. Not from the "need immediate CPR on one of my player's build choices".
I want to highlight the underlying fundamenta of the dissatisfaction of the War Priest feature. Not suggest practical fixes to anyone's individual campaign.
With that in mind, I'm sceptical of granting martial weapon proficiency. If we could go back in time and make the War Cleric the sole user of martial weapons and heavy armor, then maybe.
But as something to fix the design of War Priest (as opposed to fixing "a player of mine having an issue") it doesn't do the trick, in my opinion. All it does, after all, is add +1 DPR on your Attack action (+2 DPR while using War Priest). If the War Priest foregoes his shield, this can increase to as much as +6. Still insignificant, and still not solving the underlying issue - the focus on the Attack action.
Now you might say "but with greatweapons the path to GWM/PM feats lie open" to which I still return to the fundamental issue: as you level up, you're going to use Cast Spell. You really don't want to encourage a build choice that conflicts with this. As any good minmaxer knows, you need to pick one dimension and then optimize the hell out of that.
But in simpler terms: being good at two things does you no good if you can't do both. Much better to be stratospheric at one thing if all you can do is one thing. (Of course getting access to a second action is obviously better but let's leave that for another discussion)
You really don't want to take a feat unless you plan on using it more and more as you level up. So we're getting back to War Priest.
Regards,
Zapp
PS. Thank you for recognizing the clarity in design of the Cast Spell bonus attack suggestion.