There should be a cheaper police check option...that takes about 8weeks to process. I would find out what it costs in your area then report to this family that you are will to expense the cheaper police check that if they want the more expensive check they will have to pay the extra.
I don't believe you have to be present to drop the forms off. Confirm but I think it's just a paper with your information and your ID on it. You should be able to have your husband drop it off, maybe?
Let the other families know you need to close early to do a police check. Make sure the time you go also inconveniences the new family. I do believe having police checks is just part of this business. I agree they give a false sense of security but they are one of few measures families have to try and ensure their child is safe and we can't blame parents for that.
Also...make sure you give all families enough notice of closing early...so what ever your contract states a week or two's notice. And let the new family know that they will have to wait that long so you can properly book time off.
Hopefully, with all the above the family realizes they might be best to not make you go (they will have to pay some, they have to take time off and it will take many weeks).

