The main point, IMHO, is to make an easy to use and flavorful compound butter. It can be put on top of grilled meats, tossed in with a sautee of meat/veggies/pasta, etc.
I would finely mince some onion or shallot and garlic. Lightly sweat them, then add wine and reduce. Add some fresh herbs and the butter and melt to combine. I then cool this until lightly set, then roll it in a tube in some plastic wrap and freeze it. It can then be sliced off, like tubed cookie dough, and placed on top of grilled/baked meats, etc...
The wine and herbs add a rich, deep flavor that can be used in a variety of ways.
You won't get a buttery taste from adding butter to the dough. Even in fat-rich batters like pound cake, the difference between butter and a neutral fat is subtle - it is there, but it doesn't taste like biting into a buttered toast. And in a pizza, you can't add such amounts of fat to the dough, because it will interfere with gluten production, resulting in the wrong dough texture.
If you are prebaking the pure base at some point, putting butter on top of it will give you some buttery taste. But most pizza recipes call for immediate baking with the sauce, if not toppings, without prebaking the naked base (the exception is made for baking thick-crusted pizza in a regular, low-temp oven).
Putting butter in the sauce is also an option, but it will tend to be quite overwhelmed by the tomatoes. In general, a sour taste masks fats well, and a tomato sauce is sour.
I think that your best bet is to use some toppings which remain above the cheese, and put pieces of butter on them the moment you get the pizza outside, or, if they are crustable, in the last 5 minutes of baking. This will create small patches of melted butter with a distinctive flavour. If you do it on the cheese, it will mix with the cheese fat, and make it too greasy, so toppings above the cheese should function better.
The last option is to cheat and use butter aroma - this is what many commercial bread producers use. I wouldn't do that, because I am able to distinguish it for the real thing, but many people seem to be happy with it.
Best Answer
I actually substitute high quality olive oil for everything in my cooking now. There is a 'butter oil' which is olive oil pressed/infused with celery seed and lettuce extract which actually has a remarkably similar flavor profile. So if you live near any specialty olive oil shops that is definitely something to ask for! It's great on popcorn too. And better than all these funny hydrogenated oils.
https://unrefinedolive.com/collections/infused-olive-oils/products/butter-infused-olive-oil?variant=39727564114 This is the product I was talking about.