Roald
Would you recommend this kit over the Tamiya Fletcher Class destroyer? In the airplane world Tamiya has a reputation for great fit. Not sure if carries over to ships.
The Tamiya Fletcher (1:350) is an excellent kit and one of my recommendations for a ship beginner. Build it as it comes out of the box or add some aftermarket PE, you will meet (or come close to) the original poster's $50 price target. The kit is about 25 years old and may be a bit dated compared to Tamiya's more recent Japanese destroyer kits, such as the Yukikase or Kagero. These won't meet the OP's price point. But the Fletcher kit still fits well. About the only knock against the 350 Fletcher is that the 5-inch gunhouses are a bit underscale.
My other recommended 1:350 beginner kit is the Trumpeter USS England destroyer escort. It also has very good fit and detail. It meets the OP's pricepoint.
The Trumpeter 1:350 kit of the The Sullivans doesn't make my recommendations. Numerous problems: lack of detail or what detail is there is soft, the non-skid tiles are a raised sidewalk, the 40mm guns are sticks on boxes, the 20mm guns are about 10 feet tall.
I don't recommnend a larger subject for a ship NOOB. They are more expensive (especially when you add PE). With a large/expensive kit you may be hesitant to work on it lest you screw things up (no work, no progress, no learning). There are more repetitive assemblies which make it seem like you are not making progress. As a NOOB you want to see progress, learn, and move to another project where you can apply the things which you have learned. Do the battleship second.
I also do not recommend that you start with a smaller (1:700) kit, especially if you are adding PE. You want to teach your 15 dancing thumbs how to form the pieces and apply them. It is easier with a 1:350 piece which is 1/4 inch tall than a 1/8 inch 1:700 part. Learn how doing it larger, then transfer the muscle memory to the smaller scale.
I also do not recommend a larger scale (1:144/1:200). There are fewer subjects, they are more expensive as is any aftermarket you may want to add.