Read it twice, answer the second reading
The first reading provokes a defence. The second finds the actual request buried in it. Answer that.
Acknowledge a specific fact
“Sorry for any inconvenience” inflames, because it is obviously generic. “You were told Tuesday and it is now Friday” defuses, because it proves you read what they wrote.
Give a date, not a reassurance
People escalate because they do not know when it ends. A specific commitment — even an unwelcome one — usually stops the escalation immediately.
Do not match the register
Stay plain and factual. Every sentence you write can be screenshotted, and the calm reply always reads better in public than the clever one.
Know when to move channel
After two angry exchanges in writing, a five-minute phone call resolves what ten more emails will not. Then write a summary onto the ticket, so the agreement exists in the record.