Hello bhoseit, and welcome to the forum!
Salerno is where you are looking for but not exactly what you are looking for since cozy isn't a term I'd use. Amalfi Coast travel is only by bus or ferry and Salerno is where the ferries and buses leave from south of AC so it is the place to be. It is also where the train heads north to Pompeii and Naples as well as to all areas south like Paestum and San Lucido.
You could look into Vietri sul Mare as the last official Amalfi coast town that is connected to Salerno by train and bus but some of the AC cliff town charm. I don't know about the heart of the season but other times rentals are certainly possible and maybe month long rentals if you book early enough. You would need to bus further down the coast for the "real AC" or probably head to Salerno for ferry connections, but this probably as close as I think you can realistically get.
My only other suggestion might be Agropoli, but I mention this town having been through it but never stayed there.
It is south of Salerno by 30-45 minutes on the train It has an older historic section and more modern side and might work for you. All seaside places do some beach vacation business so it will have summer rentals but I'm sure the trade in Agropoli is nearly all Italian. It suspect will also be significantly cheaper than an anywhere on or near the Amalfi coast.
Having stayed for month in one spot in Italy last year I would ask you "How is your Italian?". If you were to head anywhere south of Salerno the need for Italian to function in a daily way is probably much higher.
This sounds awesome - hope you find what you're looking for,
=Tod