Do not focus on the chipset. You need an 8th gen processor that is supported by your motherboard BIOS. You need to ask them which processors are supported and select one of those.
Doc (not an Intel employee or contractor)
So, you need an 8th gen processor to use temporarily, as a stepping stone to your 9th gen processor. My suggestion would be to take the PC to a local PC repair shop. They will have such a processor and will be able to help you do the BIOS upgrade for a fairly nominal charge (compared to cost of another processor).
If no such shop exists and you are really stuck with purchasing another processor, I would look at whether there is a Celeron family processor that is supported (these are a lot cheaper than Core family processors). After a quick check (here: https://www.msi.com/Motherboard/support/B360-GAMING-PLUS#support-cpu), I then found a Celeron G-4930 for US$50 on Amazon (Here: https://www.amazon.com/dp/B07S8DWXTC).
Hope this helps,