I am thinking of inviting over my boyfriend while I am babysitting.
Why do you want your boyfriend to stay over?
I would love to see his reaction on finding me with another man.
It's fine to have guests over to a house you're sitting, but only if you clear them with the owner first. The homeowners put their trust in you and you shouldn't betray that responsibility to them.
Only seeing his reaction is enough.
That doesn't sound fair to your boyfriend. If you are planning such deviousness, just break up with them and spare them the pain. Stop delaying the inevitable and do it properly.