OMG some of the comments...
Don't worry OP. Shit happens. You can fix it. Sounds like it's just a short barking burst at 5am, right? 10-15 seconds? My dog used to bark constantly through the night when I first adopted him. Bottles of wine /bribes were purchased for our neighbours. They were very patient. Most people are, if you communicate with them openly and reassure them that you're working on it! :)
What actually worked with my dog, if this helps, is to teach a settle command.
Quiet just wouldn't work. My dog (working Cocker) quickly figured out that if he barked, and then was quiet when I asked, he got a treat. So he'd bark to get the treat. It doesn't always happen, and teaching quiet can be effective - it just wasn't for my dog.
Now when my dog barks, I say "settle down" and he knows to be quiet, lie down, and settle. I've also taught him "go sleepy" to let him know that he doesn't need to be on guard and can just rest and go back to sleep.
I taught "settle down" using the Relaxation Protocol during the day time. Now when I say "settle" or "settle down", he'll do so (at any time of day, with or without barking).
The reason I prefer "settle down" is that it encourages the dog to actually switch his mindset / shift into a different behaviour. Which can help, given that barking is caused by an aroused mental state (i.e. settling is the opposite of aroused).
I taught "go sleepy" just by saying it to him as he was falling asleep, and praising him for relaxing into it. Over time, he just understood what it meant.
I hope this helps! Good luck and don't stress too much!
Oh and also - depending on your sitter, most do understand that dogs are a work in progress. I'm a dog walker and sometimes do overnight sitting. I've looked after plenty of dogs who alert bark in the middle of the night. One of them used to pop off at 3am without fail, for a good half an hour. Owner pre-warned me and it was fine.