What to do if a discussion letter/paper is rejected because it's not an original research article?
The first step to take, if not already done, is to answer back to the Editor in Chief, politely explaining that the status of your submission (discussion letter) seems to have been overlooked. Stress again, politely, that the policy of the journal allow explicitly such letters, and that it would be better for everyone that they either enforce this policy (and not reject letters on the ground that they are not full research articles) or remove it from the guide to the authors. Given the answer you get, there are several possible follow-ups if your letter keeps being rejected. 

You can contact the authors and see what they have to say. In fact, it would be something to be done before any of the propositions below. They may acknowledge the mistake and publish an erratum by themselves, acknowledging you, and this would make things right in the best way. If they do not answer in a satisfactory way, at least they will have been warned and your case will be stronger.

You can appeal to an ethics committee on your field, if one exists, disclosing both your letter explaining the error, the written exchanges you have had with the journal, and the conflict of interest you spotted for the authors of the original paper. Do not make assumption, just present the fact and let the committee judge for itself.

You can try to publish your letter in another journal, in order to make the official record straight. Depending on the existing venues, you may have to add some flesh to your letter and grow it to a full paper, even if short. You are right in your principles, such mistakes should not be let unknown, and a blog post is too personal and too unofficial to make it right. People should get the information on the mistakes you spotted while using the databases usually used in your field.

But, given you are a PhD student, before you take any step I very strongly advise you to ask your advisor (or another senior researcher you can trust) about it. Depending on your field, your situation, the stature of the author the work of who you criticize, you could end up in pretty bad situation if you do not beware. I cannot tell from the information you gave, but you also have to protect yourself, and unfortunately this is not always achieved by doing the right thing.