I think I would go with Intel here and still stay under your budget. For that CPU, you will have to figure out how to update the bios to work on that board. AMD has a CPU "loaner" program, but it may be a headache.
The i3 8100 will perform better than the 2200g. The only real reason to get the 2200g is for the integrated graphics. Since you will be getting a GPU, you wont be using the integrated graphics feature. If you do decide to stick with the 2200g, get faster RAM. I would look at RAM that is 3200mhz for the 2200g.
I would go with this:
PCPartPicker part list:
https://pcpartpicker.com/list/vJ3qsZ
Price breakdown by merchant: https://pcpartpicker.com/list/vJ3qsZ/by_merchant/
CPU: Intel - Core i3-8100 3.6GHz Quad-Core Processor ($109.89 @ OutletPC)
Motherboard: Gigabyte - Z370P D3 ATX LGA1151 Motherboard ($95.89 @ OutletPC)
Memory: G.Skill - Ripjaws V Series 8GB (2 x 4GB) DDR4-3200 Memory ($103.89 @ OutletPC)
Storage: ADATA - Ultimate SU800 128GB 2.5" Solid State Drive ($44.99 @ Amazon)
Storage: Seagate - BarraCuda 1TB 3.5" 7200RPM Internal Hard Drive ($44.47 @ OutletPC)
Video Card: MSI - GeForce GTX 1050 Ti 4GB GAMING X 4G Video Card ($244.98 @ Newegg)
Total: $644.11
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2018-03-07 10:47 EST-0500