Another option in Positano to look at is:
Hotel Marincanto
Personally I thought they had the BEST location in Positano and the BEST view. We stayed and love Eden Roc but if I were to go back again I would check out the costs and rooms for Hotel Marincanto. Not endorsing the hotel in anyway I just liked this location and view and I recall it had a nice pool and outdoor spaces. From limited knowledge I had of it, seemed prices really vary for good view rooms vs. not so good rooms there though, so may be over budget for a nice room with terrace. Eden Roc was good in that all of the rooms have a terrace and the same views and is within or under your price target.
My impression of the Hotel Marincanto based on not staying there is it looked like an ideal honeymoon place.
La Sirenese while beautiful, we met some people and they were staying here so we walked back with them one evening and saw the place is definitely the chic, luxury place to stay and super convenient to the beach but I did not feel the view was as good since it is not "high" enough.
Hard to look at an online map and understand since it is a vertical town of steps what looks right next door could be a decent walk away.
Lulu mentions Eden Roc is way up and not convenient as such, I guess this depends more on what transportation methods you will be using. If arriving/departing by ferry Eden Roc is not convenient, if arriving/departing by bus, car, etc... Eden Roc is more convenient than a place lower since it is right on the main road. You can go even higher into the hills and get even higher up views but that would start to be too high up for me.
I think everyone will have different preferences, to be high and get more sweeping views or be closer to the beach. I thought position wise where the Marincanto sits offers a bit of both.
In Sorrento you will definitely get more hotel for the money, larger rooms, big lobbies, much more common areas, larger pools, areas to sunbath, etc... Space is tighter in Positano. We stayed at a nice place in Sorrento up on the cliffs with an amazing view of the Bay of Naples.
Hotel Bel Air and it was a very dreamy, honeymoon type of place that would be under your budget I am fairly certain even with a nice terrace suite.
if a perfect honeymoon involved staying in your room, at the pool or in the hotel's property and eating dinners at the hotel restaurant than the hotel Bel Air or similar hotel in Sorrento would be a better option than anything in Positano. Positano is a different feel in many ways and I preferred it greatly as a place to stay because of it's unique charm.