How do I raise labor rates without losing customers?

Announce it once, in writing, to existing customers 30 days out. 'Effective [date], our service call is $X and our hourly is $Y.' No apology, no long explanation. Businesses that apologize for raising prices lose more customers than businesses that just raise them.

You will lose 5% to 10% of the price-shopper segment. That is the goal, not the risk. Those customers were unprofitable anyway.

Same day you raise, update every quote template, invoice template, dispatch script, and website. Owners raise the rate and then techs quote the old rate for six months. Systems have to move with the price.