Thanks for reaching out about this.
I can see what you're meaning there in terms of not being able to find BlueDog on the third-party and alternative payment providers list, within the Shopify admin. Because BlueDog is a third-party payment provider outside of Shopify, we're not too familiar with it and how they can integrate onto Shopify. Usually when it comes to third-party or alternative payment processors, after you sign up with the payment provider they will give you an API key that you can input onto your Shopify store. Have you signed up with BlueDog and received an API key from them at all?
I did a bit of digging here too on my end and it looks like BlueDog integrates with a payment processor called Authorize.net. What seems to be the process here is that you would need to contact Authorize.net to create an account, and they will give you an API key. From there you can give that API key to BlueDog, who will then connect your account via that API integration.
In saying this, I also went to BlueDog's website and it looks like they have a phone number to call and the option to speak with their Support Team via a live chat. If you haven't already done so, I'd suggest getting in touch with the BlueDog team just to confirm the correct process here. From their website, they do mention that they're able to integrate onto Shopify but doesn't specify how. So reaching out to them would shed some light here on the next best steps for you.
Out of curiosity, is there a particular reason that you're wanting to use BlueDog for your payment processor? We have our very own Shopify Payments available as a payment provider if your store operates in a country that is supported. As long as your store is in line with our Shopify Payments Terms of Service, there shouldn't be any issues for you to be able to use our payment gateway on your store.
Let me know how you get on with the information I've provided you with! I'd be happy to help further if you have any other questions.
Thanks for getting back to me, @Dmash1. That totally makes sense, especially if you're already familiar and are happy with their service!
All the best with this integration, and your business!