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I assume Palakkad Tamil, which is more Malayalam than Tamil.

I'm guessing Tamil Nadu and Karnataka.

TN and KA

Now I'm curious as to the name of this dialect, please?

It sounds likely to be a language in the Tamil-Kannada family (as these are the major languages of Tamil Nadu and Karnataka respectively). There were a great number of Vijayanagara-era migrations, but sadly the South Dravidian languages have relatively less comparative linguistic study, so it's hard to say any further.

Yep, exactly. There isn't really a name for the dialect, like I said before. We call it Kannada, but it's not what you would find in Karnataka.

I've tried to do some research, but there's not very much literature and what's there is paywalled. I was even going to email one of the most prominent authors in the field of Vijayanagar-era history and migrations for a copy of her book, but found out that she'd passed away a few years prior, sadly.