I'm not an expert here, but from what I see in https://code.visualstudio.com/api/references/vscode-api, it doesn't look to me like this functionality is available through the VS Code extension API.
If you look at what VS Code is doing internally to create the chat status bar item, (search this.statusbarService.addEntry, getEntryProps, and take a glance at ChatStatusDashboard), observe that IStatusBarEntryContainer.addEntry takes an IStatusbarEntry, which can have a tooltip property which is a TooltipContent | ITooltipWithCommands, where TooltipContent is string | IMarkdownString | HTMLElement | IManagedHoverTooltipMarkdownString | IManagedHoverTooltipHTMLElement. Compare with the documented extension API StatusBarItem type definition, whose tooltip member is of type string | MarkdownString (see also IExtensionStatusBarItemService.setOrUpdateEntry).
I suppose you could try to assign an HTMLElement or TooltipWithCmmands to the tooltip on what you get from createStatusBarItem, and see if the extension facilities of VS Code reject it or not. I kind of expect it to, but during my skimming of source code, didn't see anything that immediately shows that it would. If it rejects it, I'd suggest you to raise a feature request in the VS Code issue tracker to request this as a feature.