Have you seen these kayaks yet? I am guessing you haven't been able to get them in the water and are just shopping ads.
The best bet is to actually try out kayaks in the water before buying. If you look around in the spring and summer you probably can find demos days so you can at least get a feel what you like and then look used.
But if you want to take a chance and buy a kayak first then going used is a bit safer since you probably can turn around and sell the kayak for what you paid if you don't like it.
Seems like the Malibu is outfitted somewhat for fishing by default while the Tarpon new comes in both a fishing and non-fishing model.
The Malibu is shorter, wider, and claims a higher weight capacity. Probably very stable but that will come at the cost of speed and from reading reviews they sound hard to turn. For fishing and paddling around mellow lakes and rivers it probably fine if you aren't in rush.
The Tarpon is much better known brand kayak with a lot more good reviews. The one you are looking at is longer and also narrower. Still should be a very stable kayak for fishing and dog carrying. Might be a better choice for you since you are the shorter side. I don't think you should have problems with this kayak for what it seem you want to do. If the Tarpon isn't outfitted for fishing you may need to spend a bit more to add some rod holders.
I see lots of people on sit on kayaks porting even larger dogs. Both are probably fine for that.
PFD's (Life Jackets)
Wall Mount Boat Racks