A "between centres" grinder is baasically a centre lathe with a grinder head in the toolpost. Some time ago it was realised that a better finish resulted if you freed the workpiece from being supported by the centres and fed it between two rotating wheels, one abrasive. They're usually horizontally opposed and the weight of the workpiece is also supported by a roller or two beneath.
If you haven't already, reading LTC Rolt's famous book "Tools for the Job" is recommended - a history of engineering, through its machine tools. A key argument is that the reliability, interchangeability and low cost of 20th century light engineering was the result of taking the
19th century and replacing the point cutting tools (lathes and mills) with grinding.For dowel making, then replace the wheel with an inflatable or wood and felt drum, carrying a taut abrasive band. If I did this myself, I'd use an old lathe bed (an enormously useful piece of junk I once scrounged) which already has a few two-wheel support carriages to fit onto it, then I'd make a plywood bracket up to support an old '60s B&D drill above the shaft, carrying one of my inflatable sanding drums. Make the drum height adjustable (pivot and spring-and-turnbuckle) and twist the dowel by hand.
I get narrow chair spindles turned by a guy across town, because he's a far better turner than I am. His technique is to use his left hand with a leather sailmaker's palm on it as a travelling steady, while he turns with an oval skew in his right hand and a long handle to it, tucked under his arm. They need sanding from the skew, but he can turn a very nice consistent shape for skinny spindles with a swell in the middle.