Ok so after having a Teams meeting with your support guys the answer is: You don't support the minimum tier, you won't look into why it's unstable, and your only advice is to upgrade to a General Purpose vs Burstable tier.
Given the price of the General Purpose tier, Azure is not a product I'd ever recommend to a small business. But, since I work for a corporation, I decided to test out another burstable compute size.
I moved from B1s 1 vCore 1GiB ($8.07/m) to B2s 2v Cores 4GiB ($48.84/m) and saw an immediate resolution to the issues. No longer was the CPU all over the place when under no load, but instead it is a fairly solid 2 - 3% when idle, except for a process that seems to be running for 1 hour every 8 hours which consumes 10%; no idea what this is, presumably it's an Azure process as it's not me.
Based on the simple difference of these two levels, I believe that your B1s offering for MySQL flexible server is fundamentally broken at this time, and it is frustrating that you will not look to resolve it.
Here's some graphs showing the minimum and maximum CPU usage, and the inserts / selects on the database. See if you can work out when I upgraded.