The more common way is "Podría darme una cerveza grande" (gran cerveza sound weird).
Essentially, what it means is "Podría darme una botella grande de cerveza", you know that wiht big we refer to the container.
The more informal way is "Dame (tú) una cerveza grande".
On the other hand, have in English has many meanings, two of them are eat and drink.
So, "Could I have a beer?" could mean "¿Podría beber una cerveza?", but this is too literal and we just say "¿Tiene cerveza?" or, more informal and direct, "(Tengo sed), dame una cerveza".
Saludos