Spectacular question. One thing that Nashville is known for is ecclectic, diverse, affordable homes. According to what you need you have several options at that price range.
1. Woodbine, Glenrose, Murray Hills: mostly ranch homes, finding a 2 bath may be a challenge, but 1 and 1.5 baths are commonplace. Within 5 to 8 minutes to downtown.
2. East Nashville (specifically Inglewood, Shelby Village, Rosebank, Maxwell Heights, Neil S Brown, Cleveland Park, and McFerrin Park): Expect every sort of home you can imagine from bungalows to Tudors to new construction to ranch homes. Again, if your bed and bath needs are flexible, the neighborhood has a lot to offer. Areas vary from 3 to 12 minutes from downtown.
3. Sylvan Heights: You'll find several cottages and some new construction; however, the new stuff will more than likely be above the $200K mark. 3 minutes to downtown.
4. The Nations & Charlotte Park: Lots of homes varying from ranches in Charlotte Park to tons of variety in the The Nations. 5-7 minutes to downtown
5. TSU/Fisk/Meharry: Ecclectic homes, new construction, cottages, Victorians all easily within your budget. 3-5 minutes to downtown.
6. Hope Gardens: A few options are here, some are new and some exist. You'll have to be open to the stock because it is such a small little neighborhood. walkable to downtown.
7. Villa Place: I LOVE LOVE LOVE this area and it could be your answer. There's actually two homes I can think of that I've been in that may fit the order. Some miss the "cute" factor, but it's walkable to The Gulch and Demonbruen shops. 2 minutes to downtown
8. Historic Chestnut Hill: Very very tiny neighborhood chocked full of historic, early 1900s homes. Within walking distance to the new Rolling Mill Hill. 2 minutes to downtown.
9. Woodland-in-Waverly: Limited stock, but when you find it, it's good. Location rocks with walkability to Douglas Corner. 3 minutes to downtown.
Best of luck in your search. I know it can be a struggle to find the perfect fit, but hang in there...you'll find it!