People on TFNs get paid because they are employees. If companies don't pay employees they have bigger problems than Fair Work - it's a red flag for insolvent trading.
Your husband's invoices would be treated much the same as any other invoice for a service rendered. They go through various checks and approvals before being submitted for payment. That can take time, especially when people are busy.
It's the downside of being self-employed, particularly if you only have one client.
I wouldn't send a threat for legal action unless I was genuinely intending to follow it through and prepared to wear the loss of the client.