There's data on Reloader's Nest under Handgun loads. One that gives 2290fps. Hornady lists one 40 grain .222" bullet. Actually made for the Jet. Finding 'em near you may be an issue. Shooter's Choice, in Waterloo, Ont. wants $29.95 per 100. They say they keep 'em in stock though. Bound to be somebody closer.
Everybody else makes .223" or .224" bullets. Uses a Number 6 RCBS shell holder. Same one as .38/.357.
Got the dies? Hornady, RCBS and Redding make 'em. Likely be a special order.
"...can simply lube and size .357 Mag brass..." Requires 2 case forming dies. Redding makes it. Run $63.99US each.