Assuming you are in the US, you are already well within your budget to get a Ryzen 5 2600, B450 motherboard and 16GB of ram. You also already have enough to cover the applicable taxes if your budget is $310. I'm saying you should get a $120 Ryzen 5 2600, $75 Asrock B450M Pro 4 (Micro ATX) or $80 B450 Pro 4 (ATX) and $60-70 DDR4 3000-3600Mhz 2x8GB ram kit for a total of $250-270 before taxes. You don't actually need to buy a new cooler either as the stock cooler for the Ryzen 5 2600 is sufficient to keep the CPU cool enough.
Also, just because the motherboard says it supports DDR3 2800, does not mean it will work or even be available for purchase. You are not going to find DDR3 2800 modules for cheap if you do find them and they will most certainly be used kits. They will at least cost as much as a new DDR4 3200 kit. I recently bought a 2x8GB DDR3 2400Mhz kit used for $50 and it doesn't even run stable at the rated specs on my Asrock Z97 board.