I found that if you switch to numbers, which you pointed out will allow you to choose a list and the level, you can switch back to bullets and the list and level is maintained.
That is just how InDesign works—so it's feature, not a bug. This forum automates bullets, but InDesign gives us complete control over all options.
So, use the bullets button on the Control Panel to create the first level. I'd recommend saving it as a ¶ style.
Then, select the first group of second level bullets. Hold Alt or Opt and click on the Bullets button to open Bullets and Numbering. Click Add.
Pick the desired bullet, click on Add, then Ok.
Choose the new symbol and increase the Left Indent value.
This may be more work than you were hoping for, but does this forum—or Word—give you easy access to flower bullets?