I recently bought several thousand 'processed' 5.56 Lake City cases that did not have the primer pockets reamed properly to allow consistently normal seating of the CCI 41 primers. Some of the primers appear flattened from the force required to seat the primer. If I use a modest H322 charge for training under a 77gr SMK bullet, can these cartridges be fired safely?

Did you prime them by your self?
It may be a risk for gas leakage.

Yes, I primed them myself. I could feel some of them going in hard and visually sorted the perfect ones from the questionable ones.

Gas leakage would be bad. It's not worth the risk for the price of the messed up primers. I'll deprime the questionable ones and use my primer pocket reamer to be on the safe side.

What was the result of using your primer pocket reamer, did it help? Most of these tools only cut on the bottom so if it's a diameter issue then I would suspect that the results would be minimal. In my experience, some primer pockets are definitely tighter in some manufactures brass, e.g., Norma, Lapua but loosen up a bit after a round or two. However, it sounds like yours were excessively tight. As far as my own loading goes, I always use a primer pocket reamer every time I load a pice of brass, new or used. This may be overkill but I like my primers to seat properly.

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I always use a primer pocket reamer every time I load a pice of brass, new or used. This may be overkill but I like my primers to seat properly.

That's a great idea. I'm going to adopt it after I deprime these and start over.

Bruce I bulk load 223 & 30-06 in military brass with pulled bullets & powder. From time to time I have the same problem even though I processed the pockets. The primers were over tight & somewhat flat. I set these aside once loaded & fired them separately in my bolt guns without problem. Your mileage may vary.

I've had this problem with mil brass also. Some primers run bigger than others. CCI SRP usually gave me the most problem and Fed were the best.
You might try a different brand. FWIW, I sometimes flattened the hell out of the primers getting them seated but never noticed a problem with performance.