Positano vs. Amalfi vs. Praiano

Have not stayed in any of these hotels. Do keep in mind in Italy the star rating is only based on amenities offered, it has nothing to do with level or service or other factors so don't get too caught up in the number of stars.

I walked by the Hotel Marincanto a few times and it is gorgeous and has probably the very best location in Positano for views; Positano is also the best of the 3 towns in my opinion so an obvious no brainer for me ;
However I believe I heard about that place that some of their cheaper rooms are subpar and for your timeframe and budget you may get a less desirable room. That may no longer be true, so do your own research to determine that.
Marincanto is a 4 or 5 star place I believe.
The town is tiny so I wouldn't stress too much, location wise the Albergo California is also in a great spot. I am not as fond of where the Casa Buonocore is in terms of views but then again it could be more quiet and less crowds there than the other 2. That place seems more like a smaller boutique hotel than the other 2 as well so you likely have a preference for that or not.

As to which town ; if you stay in Praiano you should get more value for your money in terms of the property but you lose out on convenience. If you want to get away and escape the crowds, seclusion in a beautiful place it is a good choice as both Positano and Amalfi are very crowded places in July and small so hard to escape those crowds.
Praiano is not remote by any means but is too far to walk to either town and there is not much in Priano itself so you are going to have to depend on taking the very crowded at that time buses more often than if you stay in one of the other twos. Or worse yet could want a car if staying in Praiano which may work well for some parts of the year but in July is not a good idea.

Positano is the prettiest of the towns and the only one with a nice beach if that matters to you.
Both Amalfi and Positano will have plenty of shops and restaurants. Praiano will have much less.

I walked from Praiano to Amalfi, but a long time ago when there was a lot less traffic. Would not be as much fun now.

I love Praiano but there is no ferry stop, so the bus is your only option.

We just got back from that area. We stayed 5 nights in Positano. We went to both Amalfi and Praiano. Amalfi was super busy. Praiano doesn't have much. I loved our stay in Positano, we were just outside the town at the Arienzo bus stop. A little quieter, AMAZING views and a very easy walk into Positano. If we had to do it over again I would not have changed a thing.

You have received some great advice and I would second it all. Positano would be my honeymoon stop ! We liked the ferry option as well as the bus to get around. I usually love the idea of staying a bit more remote, like Atrani, but you have to weigh the ease of connections if you wish to be out and about.

We stayed at Albergo California and loved the terrace views of the ocean for our breakfast each day. We were also 3 minutes from the Sponda bus stop to get around.

1. Take the bus up to Nocelle and hike the Path of the Gods into Praiano.
2. Ride the bus to Amalfi and bus up to Ravello.
3. In Ravello, take Mama Agata's cooking class - she is wonderful and the day cooking and eating overlooking the sea is spectacular !
4. Make sure you take the ferry to see the coastline. It could be from Amalfi back to your hotel in Positano.

Travel to Amalfi coast October 2017. We stayed in Amalfi at BB Pansa #2 (Ravella suite room). Advantages includes veranda with a port view and multi layered heavy doors to shut out the noise of the road below. We did not even hear the church bells in the morning. The advantage of Amalfi is public transportation hub. The buses of to Boomerang (Path of Gods trail) and Ravello. If I was going for high end stay, Ravello would be on my list. Ravello is also isolated. Note if your stay in Positano transportation will be stairs (up and down), scenic crowded bus (to stop up or down stairs), and ferry (lovely). Positano to Praiano to Amalfi by bus is about mid cliff location. If coming down from Path of Gods trail transfer is a bit higher on the cliff. Praiano stay may also include stairs. We had three excellent dinners in Amalfi. To eat outside in July you will need to pre book an eatery.

would it be too ambitious/too much work to do a couple nights in a more budget-friendly locale and a few nights in something higher-end, say, up in Ravello?

If you are looking at Ravello, you can do budget friendly and lovely at the same time. Punta Civita, Villa San Cosma and Giuliana's view are all well below your price point and offer unbeatable views. Alternatively, take a look at apartments on airbnb.